2.7 Linear Applications

Jared Eusea; Karen Perilloux; Kristina VanDusen; Lauren Johnson; Prakash Ghimire

2.7 Linear Applications

Learning Objectives

In this section you will:

Solve coin word problems.
Solve ticket and stamp word problems.
Solve a formula for a specific variable.

Solve Coin Word Problems

Imagine taking a handful of coins from your pocket or purse and placing them on your desk. How would you determine the value of that pile of coins?

If you can form a step-by-step plan for finding the total value of the coins, it will help you as you begin solving coin word problems.

One way to bring some order to the mess of coins would be to separate the coins into stacks according to their value. Quarters would go with quarters, dimes with dimes, nickels with nickels, and so on. To get the total value of all the coins, you would add the total value of each pile.

A large stack of pennies, a large stack of nickels, a shorter stack of dimes, and a stack of quarters. There are several coins in the background. — **Figure 1** To determine the total value of a stack of nickels, multiply the number of nickels times the value of one nickel.(Credit: Darren Hester via ppdigital)

How would you determine the value of each pile? Think about the dime pile—how much is it worth? If you count the number of dimes, you'll know how many you have—the number of dimes.

But this does not tell you the value of all the dimes. Say you counted 17 dimes, how much are they worth? Each dime is worth $0.10— that is the value of one dime. To find the total value of the pile of 17 dimes, multiply 17 by $0.10 to get $1.70. This is the total value of all 17 dimes.

$\begin{matrix} 17 \cdot $0.10 = $1.70 \\ number \cdot value = total value \end{matrix}$

Total Value of Coins of the Same Type

For coins of the same type, the total value can be found as follows:

$number \cdot value = total value$

where [latex]\text{number}[/latex] is the number of coins, [latex]\text{value}[/latex] is the value of each coin, and [latex]\text{total value}[/latex] is the total value of all the coins.

You could continue this process for each type of coin, and then you would know the total value of each type of coin. To get the total value of all the coins, add the total value of each type of coin.

Let's look at a specific case.

Adding Coin Values

Suppose there are 14 quarters, 17 dimes, 21 nickels, and 39 pennies. We'll make a table to organize the information – the type of coin, the number of each, and the value.

Table 1
Type	$Number$	$Value ($)$	$Total Value ($)$
Quarters	$14$	$0.25$	$3.50$
Dimes	$17$	$0.10$	$1.70$
Nickels	$21$	$0.05$	$1.05$
Pennies	$39$	$0.01$	$0.39$
			$6.64$

The total value of all the coins is $6.64.

Notice how Table 1 helped us organize all the information. Let's see how this method is used to solve a coin word problem.

Finding Number of Coins

Adalberto has $2.25 in dimes and nickels in his pocket. He has nine more nickels than dimes. How many of each type of coin does he have?

Show Solution

Step 1. Read the problem. Make sure you understand all the words and ideas.

Determine the types of coins involved.

Think about the strategy we used to find the value of the handful of coins. The first thing you need is to notice what types of coins are involved. Adalberto has dimes and nickels.

Create a table to organize the information.
- Label the columns ‘type’, ‘number’, ‘value’, ‘total value’.
- List the types of coins.
- Write in the value of each type of coin.
- Write in the total value of all the coins.

We can work this problem all in cents or in dollars. Here we will do it in dollars and put in the dollar sign ($) in the table as a reminder.

The value of a dime is $0.10 and the value of a nickel is $0.05. The total value of all the coins is $2.25.

Type	$Value ($)$	$Total Value ($)$
Dimes	$0.10$
Nickels	$0.05$
		$2.25$

Step 2. Identify what you are looking for.

We are asked to find the number of dimes and nickels Adalberto has.

Step 3. Name what you are looking for.

Use variable expressions to represent the number of each type of coin.
Multiply the number times the value to get the total value of each type of coin.

In this problem you cannot count each type of coin—that is what you are looking for—but you have a clue. There are nine more nickels than dimes. The number of nickels is nine more than the number of dimes.

Let d= number of dimes.
d+9 = number of nickels.
Fill in the "number" column to help get everything organized.

Type	$Number$	$Value ($)$	$Total Value ($)$
Dimes	$d$	$0.10$
Nickels	$d + 9$	$0.05$
			$2.25$

Now we have all the information we need from the problem!

You multiply the number times the value to get the total value of each type of coin. While you do not know the actual number, you do have an expression to represent it.

And so now multiply [latex]\text{number }\cdot\text{ value}[/latex] and write the results in the Total Value column.

Type	$Number$	$Value ($)$	$Total Value ($)$
Dimes	$d$	$0.10$	$0.10 d$
Nickels	$d + 9$	$0.05$	$0.05 (d + 9)$
			$2.25$

Step 4. Translate into an equation. Restate the problem in one sentence. Then translate into an equation.

Sum of the value of the dimes (0.10d) and value of the nickels 0.05(d+9) is total value of the coins (2.25).

Step 5. Solve the equation using good algebra techniques.

Explanation	Steps
Write the equation.	[latex]0.10d+0.05(d+9)=2.25[/latex]
Distribute.	[latex]0.10d+0.05d+.045=2.25[/latex]
Combine like terms.	[latex]0.15d+0.45=2.25[/latex]
Subtract 0.45 from each side.	[latex]0.15d=1.80[/latex]
Divide to find the number of dimes.	[latex]d=12[/latex]
The number of nickels is d + 9	[latex]d+9[/latex]
	[latex]12+9=21[/latex]

Step 6. Check.

12 dimes: [latex]112(0.10) = 1.20[/latex]

21 nickels: [latex]21(0.05)=1.05[/latex]
$2.25✓

Step 7. Answer the question.

Adalberto has twelve dimes and twenty-one nickels.

If this were a homework exercise, our work might look like this:

A possible method of solving the problem, with a table and the complete equation worked out, showing d=12 dimes.

Adalberto has twelve dimes and twenty-one nickels.

Try It

Michaela has $2.05 in dimes and nickels in her change purse. She has seven more dimes than nickels. How many coins of each type does she have?

Show Solution

Michaela has 16 dimes and 9 nickels.

Try It

Liliana has $2.10 in nickels and quarters in her backpack. She has 12 more nickels than quarters. How many coins of each type does she have?

Show Solution

Liliana has 17 nickels and 5 quarters.

How To

Solve a coin word problem.

Step 1. Read the problem. Make sure you understand all the words and ideas, and create a table to organize the information.

Step 2. Identify what you are looking for.

Step 3. Name what you are looking for. Choose a variable to represent that quantity.

Use variable expressions to represent the number of each type of coin and write them in the table.
Multiply the number times the value to get the total value of each type of coin.

Step 4. Translate into an equation. Write the equation by adding the total values of all the types of coins.

Step 5. Solve the equation using good algebra techniques.

Step 6. Check the answer in the problem and make sure it makes sense.

Step 7. Answer the question with a complete sentence.

You may find it helpful to put all the numbers into the table to make sure they check.

Table 2
Type	Number	Value ($)	Total Value

Finding Number of Coins

Maria has $2.43 in quarters and pennies in her wallet. She has twice as many pennies as quarters. How many coins of each type does she have?

Show Solution

Step 1. Read the problem. Determine the types of coins involved: We know that Maria has quarters and pennies.
Create a table to organize the information:

Label the columns: type, number, value, total value.
List the types of coins.
Write in the value of each type of coin.
Write in the total value of all the coins.

Type	$Value ($)$	$Total Value ($)$
Quarters	$0.25$
Pennies	$0.01$
		$2.43$

Step 2. Identify what you are looking for: We are looking for the number of quarters and pennies.

Step 3. Name: Represent the number of quarters and pennies using variables.:

We know Maria has twice as many pennies as quarters. The number of pennies is defined in terms of quarters.

Let [latex]q[/latex] represent the number of quarters.

Then the number of pennies is [latex]2q[/latex].

Type	$Number$	$Value ($)$	$Total Value ($)$
Quarters	$q$	$0.25$
Pennies	$2 q$	$0.01$
			$2.43$

Multiply the ‘number’ and the ‘value’ to get the ‘total value’ of each type of coin.

Type

Number

Value ($)

Total Value ($)

Quarters

q

0.25

0.25 q

Pennies

2 q

0.01

0.01 (2 q)

2.43

Step 4. Translate.
Write the equation by adding the 'total value’ of all the types of coins.

Step 5. Solve the equation.

Explanation	Steps
Write the equation.	[latex]0.25q+0.01(2q)=2.43[/latex]
Multiply.	[latex]0.25q+0.02q=2.43[/latex]
Combine like terms.	[latex].027q=2.43[/latex]
Divide by 0.27.	[latex]q=9[/latex] quarters
The number of pennies is 2q.	[latex]2q = 2 \cdot 9 = 18[/latex] pennies

Step 6. Check:

Maria has 9 quarters and 18 pennies. Does this make $2.43?

\begin{matrix} 9 quarters & 9 (0.25) & = & 2.25 \\ 18 pennies & 18 (0.01) & = & \underset{_____}{0.18} \\ Total & $2.43 ✓ \end{matrix}

Step 7. Answer the question: Maria has nine quarters and eighteen pennies.

Try It

Sumanta has $4.20 in nickels and dimes in her desk drawer. She has twice as many nickels as dimes. How many coins of each type does she have?

Show Solution

Sumanta has 42 nickels and 21 dimes.

Try It

Alison has three times as many dimes as quarters in her purse. She has $9.35 altogether. How many coins of each type does she have?

Show Solution

Alison has 51 dimes and 17 quarters.

In the next example, we'll show only the completed table—make sure you understand how to fill it in step by step.

Finding Number of Coins

Danny has $2.14 worth of pennies and nickels in his piggy bank. The number of nickels is two more than ten times the number of pennies. How many nickels and how many pennies does Danny have?

Show Solution

Step 1. Read the problem.

Determine the types of coins involved.

Pennies and nickels

Create a table. Write in the value of each type of coin.

Pennies are worth $0.01.

Nickels are worth $0.05.

Step 2. Identify what you are looking for.

the number of pennies and nickels

Step 3. Name: Represent the number of each type of coin using variables.

The number of nickels is defined in terms of the number of pennies, so start with pennies. Let p = number of pennies.

The number of nickels is two more than ten times the number of pennies.

$10 p + 2 = number of nickels$

Multiply the number and the value to get the total value of each type of coin.

Step 4. Translate.
Write the equation by adding the 'total value’ of all the types of coins.

Type	$Number$	$Value ($)$	$Total Value ($)$
pennies	$p$	$0.01$	$0.01 p$
nickels	$10 p + 2$	$0.05$	$0.05 (10 p + 2)$
			$$2.14$

Step 5. Solve the equation.

How many nickels?

[latex]\begin{array}{cc} \hfill 0.01p+0.50p+0.10 &= 2.14\hfill \\ \hfill 0.51p+0.10&=2.14\hfill \\ \hfill 0.51p &= 2.04\hfill \\ \hfill p&=4\text{ pennies}\hfill \\ \hfill 10p+2 &= 10(\textcolor{red}{4})+2\hfill \\ \hfill &= 42\text{ nickels}\hfill \end{array}[/latex]

Step 6. Check:

Is the total value of 4 pennies and 42 nickels equal to $2.14?

[latex]\begin{array}{cc} \hfill 4(0.01)+42(0.05)&\overset{?}{=} 2.14\hfill \\ \hfill 2.14 &= 2.14\,\checkmark\hfill \end{array}[/latex]

Step 7. Answer the question.

Danny has 4 pennies and 42 nickels.

Try It

Jesse has $6.55 worth of quarters and nickels in his pocket. The number of nickels is five more than two times the number of quarters. How many nickels and how many quarters does Jesse have?

Show Solution

Jesse has 41 nickels and 18 quarters.

Try It

Elaine has $7.00 in dimes and nickels in her coin jar. The number of dimes that Elaine has is seven less than three times the number of nickels. How many of each coin does Elaine have?

Show Solution

Elaine has 22 nickels and 59 dimes.

Solve Ticket and Stamp Word Problems

The strategies we used for coin problems can be easily applied to some other kinds of problems too. Problems involving tickets or stamps are very similar to coin problems, for example. Like coins, tickets and stamps have different values; so we can organize the information in tables much like we did for coin problems.

Solving Ticket Word Problems

At a school concert, the total value of tickets sold was $1,506.

Student tickets sold for $6 each and adult tickets sold for $9 each. The number of adult tickets sold was 5 less than three times the number of student tickets sold. How many student tickets and how many adult tickets were sold?

Show Solution

Step 1. Read the problem.

Determine the types of tickets involved.

There are student tickets and adult tickets.

Create a table to organize the information.

Type

Number

Value ($)

Total Value ($)

Student

[latex]6[/latex]

Adult

[latex]9[/latex]

[latex]1{,}506[/latex]

Step 2. Identify what you are looking for.

We are looking for the number of student tickets and adult tickets.

Step 3. Name. Represent the number of each type of ticket using variables.

We know the number of adult tickets sold was 5 less than three times the number of student tickets sold.

Let s be the number of student tickets. Then [latex]3s-5[/latex] is the number of adult tickets. Multiply the number times the value to get the total value of each type of ticket.

Type	$Number$	$Value ($)$	$Total Value ($)$
Student	[latex]s[/latex]	[latex]6[/latex]	[latex]6s[/latex]
Adult	[latex]3s-5[/latex]	[latex]9[/latex]	[latex]9(3s-5)[/latex]
			[latex]1{,}506[/latex]

Step 4. Translate: Write the equation by adding the total values of each type of ticket.

[latex]6s+9(3s-5)=1506[/latex]

Step 5. Solve the equation.

[latex]\begin{array}{cc}\hfill 6s+27s-45 &=1506\hfill \\ \hfill 33s-45&=1506\hfill \\ \hfill 33s&=1551\hfill \\ \hfill s&=47\text{ students}\hfill \end{array}[/latex]

Substitute to find the number of adults.

[latex]\begin{array}{cc}\hfill 3s-5 &= \text{ number of adults} \hfill \\ \hfill 3(\textcolor{red}{47})-5 &= 136\text{ adults}\hfill \end{array}[/latex]

Step 6. Check. There were 47 student tickets at $6 each and 136 adult tickets at $9 each. Is the total value $1506?

We find the total value of each type of ticket by multiplying the number of tickets times its value; we then add to get the total value of all the tickets sold.

[latex]\begin{array}{rl}\hfill 47\cdot 6 &= 282\hfill \\ \hfill 136\cdot 9 &= \underline{1224} \hfill \\ \hfill & 1506\,\checkmark \hfill \end{array}[/latex]

Step 7. Answer the question.

They sold 47 student tickets and 136 adult tickets.

Try It

The first day of a water polo tournament, the total value of tickets sold was $17,610. One-day passes sold for $20 and tournament passes sold for $30.

The number of tournament passes sold was 37 more than the number of day passes sold. How many one-day passes and how many tournament passes were sold?

Show Solution

330 one-day passes and 367 tournament passes were sold.

Try It

At the movie theater, the total value of tickets sold was $2,612.50.

Adult tickets sold for $10 each and senior/child tickets sold for $7.50 each. The number of senior/child tickets sold was 25 less than twice the number of adult tickets sold. How many senior/child tickets and how many adult tickets were sold?

Show Solution

199 senior/child tickets and 112 adult tickets were sold.

Now we'll do one where we fill in the table all at once.

Solving Stamp Problems

Monica paid $10.44 for stamps she needed to mail the invitations to her sister's baby shower. The number of 49-cent stamps was four more than twice the number of 8-cent stamps. How many 49-cent stamps and how many 8-cent stamps did Monica buy?

Show Solution

The type of stamps are 49-cent stamps and 8-cent stamps. Their names also give the value.

“The number of 49 cent stamps was four more than twice the number of 8 cent stamps.”

Let [latex]x=[/latex] number of 8-cent stamps
Let [latex]2x+4=[/latex] number of 49-cent stamps

Type	$Number$	$Value ($)$	$Total Value ($)$
49-cent stamps	[latex]2x+4[/latex]	[latex]0.49[/latex]	[latex]0.49(2x+4)[/latex]
8-cent stamps	[latex]x[/latex]	[latex]0.08[/latex]	[latex]0.08x[/latex]
			[latex]10.44[/latex]

Explanation	Steps
Write the equation from the total values.	[latex]0.49(2x+4)+0.08x=10.44[/latex]
Solve the equation.	[latex]\begin{array}{cc}\hfill 0.98x+1.96+0.08x&=10.44 \hfill \\ \hfill 1.06x+1.96 &= 10.44 \hfill \\ \hfill 1.06x&=8.48\hfill \end{array}[/latex]
Monica bought 8 eight-cent stamps.	[latex]x=8[/latex]
Find the number of 49-cent stamps she bought by evaluating [latex]2x+4[/latex] for [latex]x=8[/latex].	[latex]\begin{array}{l} 2\cdot 8 +4\hfill \\ 16+4\hfill \\ 20\hfill \end{array}[/latex]
Check.	[latex]\begin{array} \hfill 8(0.08)+20(0.49) &\overset{?}{=}10.44\hfill \\ \hfill 0.64 &\overset{?}{=}10.44\hfill \\ \hfill 10.44&=10.44\,\checkmark\hfill \end{array}[/latex]

Monica bought eight 8-cent stamps and twenty 49-cent stamps.

Try It

Eric paid $16.64 for stamps so he could mail thank you notes for his wedding gifts. The number of 49-cent stamps was eight more than twice the number of 8-cent stamps. How many 49-cent stamps and how many 8-cent stamps did Eric buy?

Show Solution

Eric bought thirty-two 49-cent stamps and twelve 8-cent stamps.

Try It

Kailee paid $14.84 for stamps. The number of 49-cent stamps was four less than three times the number of 21-cent stamps. How many 49-cent stamps and how many 21-cent stamps did Kailee buy?

Show Solution

Kailee bought twenty-six 49-cent stamps and ten 21-cent stamps.

Solve a Formula for a Specific Variable

You have used formulas throughout this book. Formulas are used in geometry, and are also very useful in the sciences and social sciences—fields such as chemistry, physics, biology, psychology, sociology, and criminal justice. Healthcare workers use formulas, too, even for something as routine as dispensing medicine. The widely used spreadsheet program Microsoft Excel^TM relies on formulas to compute its calculations. Many teachers use spreadsheets to apply formulas to compute student grades. It is important to be familiar with formulas and be able to manipulate them easily.

The formula [latex]d=rt[/latex], where [latex]d[/latex] is distance, [latex]r[/latex] is rate (or speed), and [latex]t[/latex] is time, gives the value of [latex]d[/latex] when you substitute the values of [latex]r[/latex] and [latex]t[/latex]. But what if we need to find the value of [latex]t[/latex]? We substitute the values of [latex]d[/latex] and [latex]r[/latex] and then use algebra to solve for [latex]t[/latex].

If you had to do this often, you might wonder why there isn’t a formula that gives the value of [latex]t[/latex] when you substitute in the values of [latex]d[/latex] and [latex]r[/latex]. We can get a formula like this by solving the formula [latex]d=rt[/latex] for [latex]t[/latex].

To solve a formula for a specific variable means to get that variable by itself with a coefficient of 1 on one side of the equation and all the other variables and constants on the other side. We will call this solving an equation for a specific variable. In general, this process is also called solving a literal equation. The result is another formula, made up only of variables. The formula contains letters, or literals.

Let’s try a few examples, starting with the distance, rate, and time formula we used above.

Solving a Formula for a Specific Variable

Solve the formula [latex]d=rt[/latex] for [latex]t[/latex]:

When [latex]d=520[/latex] and [latex]r=65[/latex],
In general.

Show Solution

We’ll write the solutions side-by-side so you can see that solving a formula in general uses the same steps as when we have numbers to substitute.

	When [latex]d=520[/latex] and [latex]r=65[/latex]	In general
Write the formula.	[latex]d=rt[/latex]	[latex]d=rt[/latex]
Substitute any given values.	[latex]520=65t[/latex]
Divide to isolate t.	[latex]\frac{520}{65}=\frac{65t}{65}[/latex]	[latex]\frac{d}{r}=\frac{rt}{r}[/latex]
Simplify.	[latex]\begin{array}{cc} \hfill 8 &=t\hfill \\ \hfill t&=8\hfill \end{array}[/latex]	[latex]\begin{array}{cc}\hfill \frac{d}{r} &=t\hfill \\ \hfill t &=\frac{d}{r}\hfill \end{array}[/latex]

We say the formula [latex]t=\frac{d}{r}[/latex] is solved for [latex]t[/latex].

We can use this version of the formula any time we are given the distance and rate and need to find the time.

Try It

Solve the formula [latex]d=rt[/latex] for [latex]r[/latex]:

When [latex]d=180[/latex] and [latex]t=4[/latex],
In general.

Show Solution

a. [latex]r=45[/latex]

b. [latex]r=\frac{d}{t}[/latex]

Try It

Solve the formula [latex]d=rt[/latex] for [latex]r[/latex]:

When [latex]d=780[/latex] and [latex]t=12[/latex],
In general.

Show Solution

a. [latex]r=65[/latex]

b. [latex]r=\frac{d}{t}[/latex]

We use the formula [latex]A=\frac{1}{2}bh[/latex] to find the area of a triangle when we were given the base and height. In the next example, we will solve this formula for the height.

Solving a Formula for a Specific Variable

The formula for area of a triangle is [latex]A=\frac{1}{2}bh[/latex]. Solve this formula for [latex]h[/latex]:

When [latex]A=90[/latex] and [latex]b=15[/latex],
In general.

Show Solution

	When [latex]A=90[/latex] and [latex]b=15[/latex]	In general
Write the formula.	[latex]A=\frac{1}{2}bh[/latex]	[latex]A=\frac{1}{2}bh[/latex]
Substitute any given values.	[latex]90=\frac{1}{2}\cdot 15\cdot h[/latex]
Clear the fractions.	[latex]\textcolor{red}{2}\cdot 90=\textcolor{red}{2}\cdot\frac{1}{2}\cdot 15\cdot h[/latex]	[latex]\textcolor{red}{2}\cdot A=\textcolor{red}{2}\cdot\frac{1}{2}\cdot b\cdot h[/latex]
Simplify.	[latex]180=15h[/latex]	[latex]2A=bh[/latex]
Solve for h.	[latex]12=h[/latex]	[latex]\frac{2A}{b}=h[/latex]

We can now find the height of a triangle, if we know the area and the base, by using the formula [latex]h=\frac{2A}{b}[/latex].

Try It

Use the formula [latex]A=\frac{1}{2}bh[/latex] to solve for [latex]h[/latex]:

When [latex]A=170[/latex] and [latex]b=17[/latex],
In general.

Show Solution

a. [latex]h=20[/latex]

b. [latex]h=\frac{2A}{b}[/latex]

Try It

Use the formula [latex]A=\frac{1}{2}bh[/latex] to solve for [latex]b[/latex]:

When [latex]A=62[/latex] and [latex]h=31[/latex],
In general.

Show Solution

a. [latex]b=4[/latex]

b. [latex]b=\frac{2A}{h}[/latex]

We used the formula [latex]I=Prt[/latex] to calculate simple interest, where [latex]I[/latex] is interest, [latex]P[/latex] is principal, [latex]r[/latex] is the interest rate as a decimal, and [latex]t[/latex] is time in years.

Solving a Formula for a Specific Variable

Solve the formula [latex]I=Prt[/latex] to find the principal, [latex]P[/latex]:

When [latex]I=\$5600,r=4\%,t=7[/latex],
In general.

Show Solution

	[latex]I=\$5600,r=4\%,t=7[/latex]	In general
Write the formula.	[latex]I=Prt[/latex]	[latex]I=Prt[/latex]
Substitute any given values.	[latex]5600=P(0.04)(7)[/latex]
Multiply r ⋅ t.	[latex]5600=P(0.28)[/latex]	[latex]I=P(rt)[/latex]
Divide to isolate P.	[latex]\frac{5600}{\textcolor{red}{0.28}}=\frac{P(0.28)}{\textcolor{red}{0.28}}[/latex]	[latex]\frac{I}{\textcolor{red}{rt}}=\frac{P(rt)}{\textcolor{red}{rt}}[/latex]
Simplify.	[latex]20{,}000=P[/latex]	[latex]\frac{I}{rt}=P[/latex]
State the answer.	The principal is $20,000.	[latex]P=\frac{I}{rt}[/latex]

Try It

Use the formula [latex]I=Prt[/latex]. Find [latex]t[/latex]:

When [latex]I=\$2160, r=6\%, P=\$12{,}000,[/latex]
In general.

Show Solution

a. [latex]t=3[/latex]

b. [latex]t=\frac{I}{Pr}[/latex]

Try It

Use the formula [latex]I=Prt[/latex]. Find [latex]r[/latex]:

When [latex]I=\$5400,P=\$9000, t=5,[/latex]
In general.

Show Solution

a.[latex]r=0.12[/latex]

b. [latex]r=\frac{I}{Pt}[/latex]

Mathispower4u. Solving Literal Equations - Part 1 (L7.1). YouTube. https://www.youtube.com/watch?v=OgEg5Rfp5cw

Mathispower4u. Solving Literal Equations - Part 2 (L7.1). YouTube. https://www.youtube.com/watch?v=Kjn1QP5saUg

Later in this class, and in future algebra classes, you’ll encounter equations that relate two variables, usually [latex]x[/latex] and [latex]y[/latex]. You might be given an equation that is solved for [latex]y[/latex] and need to solve it for [latex]x[/latex] or vice versa. In the following example, we’re given an equation with both [latex]x[/latex] and [latex]y[/latex] on the same side and we’ll solve it for [latex]y[/latex].

To do this, we will follow the same steps that we used to solve a formula for a specific variable.

Solving a Formula for a Specific Variable

Solve the formula [latex]3x+2y=18[/latex] for [latex]y[/latex]:

When [latex]x=4[/latex],
In general.

Show Solution

	When [latex]x=4[/latex]	In general
Write the equation.	[latex]3x+2y=18[/latex]	[latex]3x+2y=18[/latex]
Substitute any given values.	[latex]3(4)+2y=18[/latex]
Simplify if possible.	[latex]12+2y=18[/latex]
Subtract to isolate the [latex]y[/latex]-term.	[latex]12\textcolor{red}{-12}+2y=18\textcolor{red}{-12}[/latex]	[latex]3x\textcolor{red}{-3x}+2y=18\textcolor{red}{-3x}[/latex]
Simplify.	[latex]2y=6[/latex]	[latex]2y=18-3x[/latex]
Divide.	[latex]\frac{2y}{\textcolor{red}{2}}=\frac{6}{\textcolor{red}{2}}[/latex]	[latex]\frac{2y}{\textcolor{red}{2}}=\frac{18-3x}{\textcolor{red}{2}}[/latex]
Simplify.	[latex]y=3[/latex]	[latex]y=\frac{18-3x}{2}[/latex]

Try It

Solve the formula [latex]3x+4y=10[/latex] for [latex]y[/latex]:

When [latex]x=2[/latex],
In general.

Show Solution

a. [latex]y=1[/latex]

b.[latex]y=\frac{10-3x}{4}[/latex]

Try It

Solve the formula [latex]5x+2y=18[/latex] for [latex]y[/latex]:

When [latex]x=4[/latex],
In general.

Show Solution

a. [latex]y=-1[/latex]

b. [latex]y=\frac{18-5x}{2}[/latex]

In the previous examples, we used the numbers in part (a) as a guide to solving in general in part (b). Do you think you’re ready to solve a formula in general without using numbers as a guide?

Solving a Formula for a Specific Variable

Solve the formula [latex]P=a+b+c[/latex] for [latex]a[/latex].

Show Solution

We will isolate [latex]a[/latex] on one side of the equation.

We will isolate [latex]a[/latex] on one side of the equation.
Write the equation.	[latex]P=a+b+c[/latex]
Subtract [latex]b[/latex] and [latex]c[/latex] from both sides to isolate [latex]a[/latex].	[latex]P\textcolor{red}{-b-c}=a+b+c\textcolor{red}{-b-c}[/latex]
Simplify.	[latex]P-b-c=a[/latex]

So, [latex]a=P-b-c[/latex].

Try It

Solve the formula [latex]P=a+b+c[/latex] for [latex]b[/latex].

Show Solution

[latex]b=P-a-c[/latex]

Try It

Solve the formula [latex]P=a+b+c[/latex] for [latex]c[/latex].

Show Solution

[latex]c=P-a-b[/latex]

Solving a Formula for a Specific Variable

Solve the equation [latex]3x+y=10[/latex] for [latex]y[/latex].

Show Solution

We will isolate [latex]y[/latex] on one side of the equation.

We will isolate [latex]y[/latex] on one side of the equation.
Write the equation.	[latex]3x+y=10[/latex]
Subtract [latex]3x[/latex] from both sides to isolate [latex]y[/latex].	[latex]3x\textcolor{red}{-3x}+y=10\textcolor{red}{-3x}[/latex]
Simplify.	[latex]y=10-3x[/latex]

Try It

Solve the formula [latex]7x+y=11[/latex] for [latex]y[/latex].

Show Solution

[latex]y=-7x+11[/latex]

Try It

Solve the formula [latex]11x+y=8[/latex] for [latex]y[/latex].

Show Solution

[latex]y=-11x+8[/latex]

Solving a Formula for a Specific Variable

Solve the equation [latex]6x+5y=13[/latex] for [latex]y[/latex].

Show Solution

We will isolate [latex]y[/latex] on one side of the equation.

We will isolate [latex]y[/latex] on one side of the equation.
Write the equation.	[latex]6x+5y=13[/latex]
Subtract to isolate the term with [latex]y[/latex].	[latex]6x+5y\textcolor{red}{-6x}=13\textcolor{red}{-6x}[/latex]
Simplify.	[latex]5y=13-6x[/latex]
Divide by 5 to make the coefficient 1.	[latex]\frac{5y}{\textcolor{red}{5}}=\frac{13-6x}{\textcolor{red}{5}}[/latex]
Simplify.	[latex]y=\frac{13-6x}{5}[/latex]

Try It

Solve the formula [latex]4x+7y=9[/latex] for [latex]y[/latex].

Show Solution

[latex]y=\frac{9-4x}{7}[/latex]

Try It

Solve the formula [latex]5x+8y=1[/latex] for [latex]y[/latex].

Show Solution

[latex]y=\frac{1-5x}{8}[/latex]

Key Concepts

Finding the Total Value for Coins of the Same Type

For coins of the same type, the total value can be found as follows:

[latex]\text{number} \cdot \text{value} = \text{total value}[/latex]

where number is the number of coins, value is the value of each coin, and total value is the total value of all the coins.

Solve a Coin Word Problem

Step 1. Read the problem. Make sure you understand all the words and ideas, and create a table to organize the information.
Step 2. Identify what you are looking for.
Step 3. Name what you are looking for. Choose a variable to represent that quantity.
- Use variable expressions to represent the number of each type of coin and write them in the table.
- Multiply the number times the value to get the total value of each type of coin.
Step 4. Translate into an equation. Write the equation by adding the total values of all the types of coins.
Step 5. Solve the equation using good algebra techniques.
Step 6. Check the answer in the problem and make sure it makes sense.
Step 7. Answer the question with a complete sentence.

Section Exercises

Solve Coin Word Problems

In the following exercises, solve the coin word problems.

1. Jaime has $$2.60$ in dimes and nickels. The number of dimes is $14$ more than the number of nickels. How many of each coin does he have?

Show Solution

8 nickels, 22 dimes

2. Lee has $$1.75$ in dimes and nickels. The number of nickels is $11$ more than the number of dimes. How many of each coin does he have?

3. Ngo has a collection of dimes and quarters with a total value of $$3.50 .$ The number of dimes is $7$ more than the number of quarters. How many of each coin does he have?

Show Solution

15 dimes, 8 quarters

4. Connor has a collection of dimes and quarters with a total value of $$6.30 .$ The number of dimes is $14$ more than the number of quarters. How many of each coin does he have?

5. Carolyn has $$2.55$ in her purse in nickels and dimes. The number of nickels is $9$ less than three times the number of dimes. Find the number of each type of coin.

Show Solution

12 dimes, 27 nickels

6. Julio has $$2.75$ 5 in his pocket in nickels and dimes. The number of dimes is $10$ less than twice the number of nickels. Find the number of each type of coin.

7. Chi has $$11.30$ in dimes and quarters. The number of dimes is $3$ more than three times the number of quarters. How many dimes and nickels does Chi have?

Show Solution

63 dimes, 20 quarters

8. Tyler has $$9.70$ n dimes and quarters. The number of quarters is $8$ more than four times the number of dimes. How many of each coin does he have?

9. A cash box of $$1$ and $$5$ bills is worth $$45 .$ The number of $$1$ bills is $3$ more than the number of $$5$ bills. How many of each bill does it contain?

Show Solution

10 of the $1 bills, 7 of the $5 bills

10. Joe's wallet contains $$1$ and $$5$ bills worth $$47 .$ The number of $$1$ bills is $5$ more than the number of $$5$ bills. How many of each bill does he have?

11. In a cash drawer there is $$125$ in $$5$ and $$10$ bills. The number of $$10$ bills is twice the number of $$5$ bills. How many of each are in the drawer?

Show Solution

10 of the $10 bills, 5 of the $5 bills

12. John has $$175$ in $$5$ and $$10$ bills in his drawer. The number of $$5$ bills is three times the number of $$10$ bills. How many of each are in the drawer?

13. Mukul has $$3.75$ in quarters, dimes and nickels in his pocket. He has five more dimes than quarters and nine more nickels than quarters. How many of each coin are in his pocket?

Show Solution

16 nickels, 12 dimes, 7 quarters

14. Vina has $$4.70$ in quarters, dimes and nickels in her purse. She has eight more dimes than quarters and six more nickels than quarters. How many of each coin are in her purse?

Solve Ticket and Stamp Word Problems

In the following exercises, solve the ticket and stamp word problems.

15. The play took in $$550$ one night. The number of $8 adult tickets was $10$ less than twice the number of $$5$ child tickets. How many of each ticket were sold?

Show Solution

30 child tickets, 50 adult tickets

16. If the number of $$8$ child tickets is seventeen less than three times the number of $$12$ adult tickets and the theater took in $$584,$ how many of each ticket were sold?

17. The movie theater took in $$1,220$ one Monday night. The number of $$7$ child tickets was ten more than twice the number of $$9$ adult tickets. How many of each were sold?

Show Solution

110 child tickets, 50 adult tickets

18. The ball game took in $$1,340$ one Saturday. The number of $$12$ adult tickets was $15$ more than twice the number of $$5$ child tickets. How many of each were sold?

19. Julie went to the post office and bought both $$0.49$ stamps and $$0.34$ postcards for her office's bills She spent $$62.60 .$ The number of stamps was $20$ more than twice the number of postcards. How many of each did she buy?

Show Solution

40 postcards, 100 stamps

20. Before he left for college out of state, Jason went to the post office and bought both $$0.49$ stamps and $$0.34$ postcards and spent $$12.52 .$ The number of stamps was $4$ more than twice the number of postcards. How many of each did he buy?

21. Maria spent $$16.80$ at the post office. She bought three times as many $$0.49$ stamps as $$0.21$ stamps. How many of each did she buy?

Show Solution

30 at 49 cents, 10 at 21 cents

22. Hector spent $$43.40$ at the post office. He bought four times as many $$0.49$ stamps as $$0.21$ stamps. How many of each did he buy?

23. Hilda has $$210$ worth of $$10$ and $$12$ stock shares. The numbers of $$10$ shares is $5$ more than twice the number of $$12$ shares. How many of each does she have?

Show Solution

15 at $10 shares, 5 at $12 shares

24. Mario invested $$475$ in $$45$ and $$25$ stock shares. The number of $$25$ shares was $5$ less than three times the number of $$45$ shares. How many of each type of share did he buy?

Solve a Formula for a Specific Variable

For the following exercises, use the given formulas to answer the questions.

25. [latex]A=P\left(1+rt\right)[/latex]is used to find the principal amount [latex]P[/latex] deposited, earning [latex]r[/latex]% interest, for [latex]t[/latex] years. Use this to find what principal amount [latex]P[/latex] David invested at a 3% rate for 20 years if [latex]A=\text{\$}8{,}000.[/latex]

Show Solution
[latex]\$5000[/latex]

26. The formula [latex]F=\frac{m{v}^{2}}{R}[/latex] relates centripetal force ([latex]F[/latex]), velocity ([latex]v[/latex]), mass ([latex]m[/latex]), and distance ([latex]R[/latex]). Find [latex]R[/latex] when [latex]m=45,v=7,[/latex] and [latex]F=245.[/latex]

27. [latex]F=ma[/latex] indicates that force ([latex]F[/latex]) equals mass ([latex]m[/latex]) times acceleration ([latex]a[/latex]). Find the acceleration of a mass of 50 kg if a force of 12 N is exerted on it.

Show Solution
[latex]0.24 \text{m/s}^2[/latex]

28. [latex]\text{Sum}=\frac{1}{1-r}[/latex] is the formula for an infinite series sum. If the sum is 5, find [latex]r.[/latex]

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to answer a question.

29. Solve for [latex]W:\,\,P=2L+2W[/latex].

Show Solution
[latex]W=\frac{W-2L}{2}[/latex]

30. Use the formula from the previous question to find the width, [latex]W[/latex], of a rectangle whose length is 15 and whose perimeter is 58.

31. Solve for [latex]f:\,\,\frac{1}{p}+\frac{1}{q}=\frac{1}{f}[/latex]

Show Solution
[latex]f=\frac{pq}{p+q}[/latex]

32. Use the formula from the previous question to find [latex]f[/latex] when [latex]p=8\text{and }q=13.[/latex]

33. Solve for [latex]m[/latex] in the slope-intercept formula: [latex]y=mx+b.[/latex]

Show Solution
[latex]m=\frac{y-b}{x}[/latex]

34. Use the formula from the previous question to find [latex]m[/latex] when the coordinates of the point are [latex]\left(4,7\right)[/latex] and [latex]b=12.[/latex]

35. The area of a trapezoid is given by [latex]A=\frac{1}{2}h\left({b}_{1}+{b}_{2}\right).[/latex] Use the formula to find the area of a trapezoid with [latex]h=6,\text{ }{b}_{1}=14,\text{ and }{b}_{2}=8.[/latex]

Show Solution
[latex]66[/latex]

36. Solve for [latex]h[/latex]: [latex]A=\frac{1}{2}h\left({b}_{1}+{b}_{2}\right)[/latex]

37. Use the formula from the previous question to find the height of a trapezoid with [latex]A=150,\text{ }{b}_{1}=19,\text{ and }{b}_{2}=11.[/latex]

Show Solution
[latex]10[/latex]

38. Find the dimensions of an American football field. The length is 200 ft more than the width, and the perimeter is 1,040 ft. Find the length and width. Use the perimeter formula [latex]P=2L+2W.[/latex]

39. Distance equals rate times time, [latex]d=rt.[/latex] Find the distance Tom travels if he is moving at a rate of 55 mi/h for 3.5 h.

Show Solution
[latex]192.5\text{ miles}[/latex]

40. Using the formula in the previous exercise, find the distance that Susan travels if she is moving at a rate of 60 mi/h for 6.75 h.

41. What is the total distance that two people travel in 3 h if one of them is riding a bike at 15 mi/h and the other is walking at 3 mi/h?

Show Solution
[latex]54\text{ miles}[/latex]

42. If the area model for a triangle is [latex]A=\frac{1}{2}bh,[/latex] find the area of a triangle with a height of 16 in. and a base of 11 in.

43. Solve for [latex]h[/latex]: [latex]A=\frac{1}{2}bh[/latex]

Show Solution
[latex]h=\frac{2A}{b}[/latex]

44. Use the formula from the previous question to find the height to the nearest tenth of a triangle with a base of 15 and an area of 215.

45. The volume formula for a cylinder is [latex]V=\pi {r}^{2}h.[/latex] Using the symbol [latex]\pi[/latex] in your answer, find the volume of a cylinder with a radius [latex]r[/latex] of 4 cm and a height [latex]h[/latex] of 14 cm.

Show Solution
[latex]224\pi \text{ cm}^2[/latex]

46. Solve for [latex]h[/latex]:[latex]V=\pi {r}^{2}h[/latex]

47. Use the formula from the previous question to find the height of a cylinder with a radius of 8 and a volume of [latex]16\pi[/latex].

Show Solution
[latex]0.25[/latex]

48. Solve for [latex]r[/latex]: [latex]V=\pi {r}^{2}h[/latex]. Then use this formula to find the radius of a cylinder with a height of 36 and a volume of [latex]324\pi .[/latex]

49. The formula for the circumference of a circle is [latex]C=2\pi r.[/latex] Find the circumference of a circle with a diameter of 12 in. (diameter = [latex]2r[/latex]). Use the symbol [latex]\pi[/latex] in your final answer.

Show Solution

[latex]C=12\pi[/latex]

50. Solve the formula from the previous question for [latex]\pi.[/latex] Notice why [latex]\pi[/latex] is sometimes defined as the ratio of the circumference to its diameter.

In the following exercises, use the formula [latex]d=rt[/latex].

51. Solve for [latex]t[/latex]:

When [latex]d=350[/latex] and [latex]r=70[/latex],
In general.

Show Solution
a. [latex]t=5[/latex]
b. [latex]t=\frac{d}{r}[/latex]

52. Solve for [latex]t[/latex]:

When [latex]d=240[/latex] and [latex]r=60[/latex],
In general.

53. Solve for [latex]t[/latex]:

When [latex]d=510[/latex] and [latex]r=60[/latex],
In general.

Show Solution
a. [latex]t=8.5[/latex]
b. [latex]t=\frac{d}{r}[/latex]

54. Solve for [latex]t[/latex]:

When [latex]d=175[/latex] and [latex]r=50[/latex],
In general.

55. Solve for [latex]r[/latex]:

When [latex]d=204[/latex] and [latex]t=3[/latex],
In general.

Show Solution
a. [latex]r=68[/latex]
b. [latex]r=\frac{d}{t}[/latex]

56. Solve for [latex]r[/latex]:

When [latex]d=420[/latex] and [latex]t=6[/latex],
In general.

57. Solve for [latex]r[/latex]:

When [latex]d=160[/latex] and [latex]t=2.5[/latex],
In general.

Show Solution
a. [latex]r=64[/latex]
b. [latex]r=\frac{d}{t}[/latex]

58. Solve for [latex]r[/latex]:

When [latex]d=180[/latex] and [latex]t=4.5[/latex],
In general.

In the following exercises, use the formula [latex]A=\frac{1}{2}bh[/latex].

59. Solve for [latex]b[/latex]:

When [latex]A=126[/latex] and [latex]h=18[/latex],
In general.

Show Solution
a. [latex]b=14[/latex]
b. [latex]b=\frac{2A}{h}[/latex]

60. Solve for [latex]h[/latex]:

When [latex]A=176[/latex] and [latex]b=22[/latex],
In general.

61. Solve for [latex]h[/latex]:

When [latex]A=375[/latex] and [latex]b=25[/latex],
In general.

Show Solution
a. [latex]h=30[/latex]
b. [latex]h=\frac{2A}{b}[/latex]

62. Solve for [latex]b[/latex]:

When [latex]A=65[/latex] and [latex]h=13[/latex],
In general.

In the following exercises, use the formula [latex]I=Prt[/latex].

63. Solve for the principal, [latex]P[/latex]:

When [latex]I=\$5480,r=4\%,[/latex] and [latex]t=7[/latex],
In general.

Show Solution
a. [latex]P=\$19{,}571.43[/latex]
b. [latex]P=\frac{I}{rt}[/latex]

64. Solve for the principal, [latex]P[/latex]:

When [latex]I=\$3950,r=6\%,[/latex] and [latex]t=5[/latex],
In general.

65. Solve for the time, [latex]t[/latex]:

When [latex]I=\$2376,P=\$9000,[/latex] and [latex]r=4.4\%[/latex],
In general.

Show Solution
a. [latex]t=6[/latex]
b. [latex]t=\frac{I}{Pr}[/latex]

66. Solve for the time, [latex]t[/latex]:

When [latex]I=\$624,P=\$6000,[/latex] and [latex]r=5.2\%[/latex],
In general.

In the following exercises, solve.

67. Solve the formula [latex]2x+3y=12[/latex] for [latex]y[/latex]:

When [latex]x=3[/latex],
In general.

Show Solution
a. [latex]y=2[/latex]
b. [latex]y=\frac{12-2x}{3}[/latex]

68. Solve the formula [latex]5x+2y=10[/latex] for [latex]y[/latex]:

When [latex]x=4[/latex],
In general.

69. Solve the formula [latex]3x+y=7[/latex] for [latex]y[/latex]:

When [latex]x=-2[/latex],
In general.

Show Solution
a. [latex]y=13[/latex]
b. [latex]y=7-3x[/latex]

70. Solve the formula [latex]4x+y=5[/latex] for [latex]y[/latex]:

When [latex]x=-3[/latex],
In general.

71. Solve [latex]a+b=90[/latex] for [latex]b[/latex].

Show Solution
[latex]b=90-a[/latex]

72. Solve [latex]a+b=90[/latex] for [latex]a[/latex].

73. Solve [latex]180=a+b+c[/latex] for [latex]a[/latex].

Show Solution
[latex]a=180-b-c[/latex]

74. Solve [latex]180=a+b+c[/latex] for [latex]c[/latex].

75. Solve the formula [latex]8x+y=15[/latex] for [latex]y[/latex].

Show Solution
[latex]y=-8x+15[/latex]

76. Solve the formula [latex]9x+y=13[/latex] for [latex]y[/latex].

77. Solve the formula [latex]-4x+y=-6[/latex] for [latex]y[/latex].

Show Solution
[latex]y=4x-6[/latex]

78. Solve the formula [latex]-5x+y=-1[/latex] for [latex]y[/latex].

79. Solve the formula [latex]4x+3y=7[/latex] for [latex]y[/latex].

Show Solution
[latex]y=\frac{7-4x}{3}[/latex]

80. Solve the formula [latex]3x+2y=11[/latex] for [latex]y[/latex].

81. Solve the formula [latex]x-y=-4[/latex] for [latex]y[/latex].

Show Solution
[latex]y=x+4[/latex]

82. Solve the formula [latex]x-y=-3[/latex] for [latex]y[/latex].

83. Solve the formula [latex]P=2L+2W[/latex] for [latex]L[/latex].

Show Solution
[latex]L=\frac{P-2W}{2}[/latex]

84. Solve the formula [latex]P=2L+2W[/latex] for [latex]W[/latex].

85. Solve the formula [latex]C=\pi d[/latex] for [latex]d[/latex].

Show Solution
[latex]d=\frac{C}{\pi}[/latex]

86. Solve the formula [latex]V=LWH[/latex] for [latex]L[/latex].

87. Solve the formula [latex]V=LWH[/latex] for [latex]H[/latex].

Show Solution
[latex]H=\frac{V}{LW}[/latex]

Everyday Math

88. Parent Volunteer Laurie was completing the treasurer's report for her son's Boy Scout troop at the end of the school year. She didn't remember how many boys had paid the $24 full-year registration fee and how many had paid a $16 partial-year fee. She knew that the number of boys who paid for a full-year was ten more than the number who paid for a partial-year. If $400 was collected for all the registrations, how many boys had paid the full-year fee and how many had paid the partial-year fee?

89. Parent Volunteer As the treasurer of her daughter's Girl Scout troop, Laney collected money for some girls and adults to go to a 3-day camp. Each girl paid $75 and each adult paid $30. The total amount of money collected for camp was $765. If the number of girls is three times the number of adults, how many girls and how many adults paid for camp?

Show Solution

9 girls, 3 adults

90. Converting temperature While on a tour in Greece, Tatyana saw that the temperature was 40° Celsius. Solve for [latex]F[/latex] in the formula [latex]C=\frac{5}{9}(F-32)[/latex] to find the temperature in Fahrenheit.

91. Converting temperature Yon was visiting the United States and he saw that the temperature in Seattle was 50° Fahrenheit. Solve for [latex]C[/latex] in the formula [latex]F=\frac{9}{5}C+32[/latex] to find the temperature in Celsius.

Show Solution
10° Celsius

Writing Exercises

92. Suppose you have $6$ quarters, $9$ dimes, and $4$ pennies. Explain how you find the total value of all the coins.

93. Do you find it helpful to use a table when solving coin problems? Why or why not?

94. In the table used to solve coin problems, one column is labeled “number” and another column is labeled ‘“value.” What is the difference between the number and the value?

95. What similarities and differences did you see between solving the coin problems and the ticket and stamp problems?

96. Solve the equation [latex]2x+3y=6[/latex] for [latex]y[/latex]:

When [latex]x=-3[/latex],
In general.
Which solution is easier for you? Explain why.

97. Solve the equation [latex]5x-2y=10[/latex] for [latex]x[/latex]:

When [latex]y=10[/latex],
In general.
Which solution is easier for you? Explain why.

Show Solution
a. [latex]x=6[/latex]
b. [latex]x=\frac{2y+10}{5}[/latex]

Glossary

area: in square units, the area formula used in this section is used to find the area of any two-dimensional rectangular region: [latex]A=LW[/latex]

perimeter: in linear units, the perimeter formula is used to find the linear measurement, or outside length and width, around a two-dimensional regular object; for a rectangle: [latex]P=2L+2W[/latex]

volume: in cubic units, the volume measurement includes length, width, and depth: [latex]V=LWH[/latex]

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Solve Coin Word Problems

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How To

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Solve Ticket and Stamp Word Problems

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Solve a Formula for a Specific Variable

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Finding the Total Value for Coins of the Same Type

Solve a Coin Word Problem

Solve Coin Word Problems

Solve Ticket and Stamp Word Problems

Solve a Formula for a Specific Variable

Everyday Math

Writing Exercises

96. Solve the equation [latex]2x+3y=6[/latex] for [latex]y[/latex]:

Glossary

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