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2.1 Linear Equations in One Variable

Cynthia Singleton; Ginny Bradley; Karen Perilloux; and Prakash Ghimire

Learning Objectives

In this section you will:

  • Solve equations in one variable.
  • Solve a rational equation.

 

Caroline is a full-time college student planning a spring break vacation. To earn enough money for the trip, she has taken a part-time job at the local bank that pays $15.00/hr, and she opened a savings account with an initial deposit of $400 on January 15. She arranged for direct deposit of her payroll checks. If spring break begins March 20 and the trip will cost approximately $2,500, how many hours will she have to work to earn enough to pay for her vacation? If she can only work 4 hours per day, how many days per week will she have to work? How many weeks will it take? In this section, we will investigate problems like this and others, which generate graphs like the line in Figure 1.

Coordinate plane where the x-axis ranges from 0 to 200 in intervals of 20 and the y-axis ranges from 0 to 3,000 in intervals of 500. The x-axis is labeled Hours Worked and the y-axis is labeled Savings Account Balance. A linear function is plotted with a y-intercept of 400 with a slope of 15. A dotted horizontal line extends from the point (0,2500).
Figure 1

Solving Linear Equations in One Variable

A linear equation is an equation of a straight line, written in one variable. The only power of the variable is 1. Linear equations in one variable may take the form[latex],ax+b=0,[/latex]and are solved using basic algebraic operations.

We begin by classifying linear equations in one variable as one of three types: identity, conditional, or inconsistent. An identity equation is true for all values of the variable. Here is an example of an identity equation.

[latex]3x=2x+x[/latex]

The solution set consists of all values that make the equation true. For this equation, the solution set is all real numbers because any real number substituted for[latex],x,[/latex]will make the equation true.

A conditional equation is true for only some values of the variable. For example, if we are to solve the equation[latex],5x+2=3x-6,[/latex] we have the following:

[latex]begin{array}{ccc}hfill 5x+2& =& 3x-6hfill \ hfill 2x& =& -8hfill \ hfill x& =& -4hfill end{array}[/latex]

The solution set consists of one number:[latex],left{-4right}.,[/latex]It is the only solution, and therefore, we have solved a conditional equation.

An inconsistent equation results in a false statement. For example, if we are to solve[latex],5x-15=5left(x-4right),[/latex] we have the following:

[latex]begin{array}{cccc}hfill 5x-15& =& 5x-20hfill & \ hfill 5x-15-5x& =& 5x-20-5xhfill & phantom{rule{2em}{0ex}}text{Subtract }5xtext{ from both sides}.hfill \ hfill -15& ne & -20hfill & phantom{rule{2em}{0ex}}text{False statement}hfill end{array}[/latex]

Indeed,[latex],-15ne ,-20.,[/latex]There is no solution because this is an inconsistent equation.

Solving linear equations in one variable involves the fundamental properties of equality and basic algebraic operations. A brief review of those operations follows.

Linear Equation in One Variable

A linear equation in one variable can be written in the form

[latex]ax+b=0[/latex]

where a and b are real numbers,[latex],ane 0.[/latex]

How To

Given a linear equation in one variable, use algebra to solve it.

The following steps are used to manipulate an equation and isolate the unknown variable, so that the last line reads[latex],x=[/latex]________ if x is the unknown. There is no set order, as the steps used depend on what is given:

  1. We may add, subtract, multiply, or divide an equation by a number or an expression as long as we do the same thing to both sides of the equal sign. Note that we cannot divide by zero.
  2. Apply the distributive property as needed:[latex],aleft(b+cright)=ab+ac.[/latex]
  3. Isolate the variable on one side of the equation.
  4. When the variable is multiplied by a coefficient in the final stage, multiply both sides of the equation by the reciprocal of the coefficient.

Examples

Solving an Equation in One Variable

Solve the following equation:[latex],2x+7=19.[/latex]

Show Solution

This equation can be written in the form[latex],ax+b=0,[/latex]by subtracting[latex],19,[/latex]from both sides. However, we may proceed to solve the equation in its original form by performing algebraic operations.

[latex]begin{array}{cccc}hfill 2x+7& =& 19hfill & \ hfill 2x& =& 12hfill & phantom{rule{2em}{0ex}}text{Subtract 7 from both sides}text{.}hfill \ hfill x& =& 6hfill & phantom{rule{2em}{0ex}}text{Multiply both sides by }frac{1}{2}text{ or divide by 2}text{.}hfill end{array}[/latex]

The solution is 6

 

Try It

Solve the linear equation in one variable:[latex],2x+1=-9.[/latex]

Show Solution

[latex]x=-5[/latex]

Solving an Equation Algebraically When the Variable Appears on Both Sides

Solve the following equation: [latex]4(x-3)+12=15-5(x+6)[/latex]

Show Solution

Apply standard algebraic properties.

[latex]4(x-3)+12=15-5(x+6)[/latex]
[latex]4x-12+12=15-5x-30[/latex] Apply the distributive property.
              [latex]4x=-15-5x[/latex] Combine like terms.
              [latex]9x=-15[/latex] Place x -terms on one side and simplify.
[latex]x=- dfrac{15}{9}[/latex] Multiply both sides by [latex]dfrac{1}{9}[/latex], the reciprocal of 9.
 [latex]x=- dfrac{5}{3}[/latex] Simplify the fraction.

Analysis

This problem requires the distributive property to be applied twice, and then the properties of algebra are used to reach the final line,[latex],x=-frac{5}{3}.[/latex]

 

Try It

Solve the equation in one variable:[latex],-2left(3x-1right)+x=14-x.[/latex]

Show Solution

[latex]x=-3[/latex]

https://youtube.com/watch?v=CTTRs0oPsNI%26%2391%3B

Key Concepts

  • We can solve linear equations in one variable in the form[latex],ax+b=0,[/latex]using standard algebraic properties.
  • A rational expression is a quotient of two polynomials. We use the LCD to clear the fractions from an equation.
  • All solutions to a rational equation should be verified within the original equation to avoid an undefined term, or zero in the denominator.

Section Exercises

Verbal

  1. What does it mean when we say that a linear equation is inconsistent?
  2. When solving the following equation:  [latex]frac{2}{x-5}=frac{4}{x+1}[/latex] explain why we must exclude[latex],x=5,[/latex]and[latex],x=-1,[/latex]as possible solutions from the solution set.
Show Solution

If we insert either value into the equation, they make an expression in the equation undefined (zero in the denominator).

Algebraic

For the following exercises, solve the equation for[latex],x.[/latex]

  1. [latex]7x+2=3x-9[/latex]
  2. [latex]4x-3=5[/latex]
Show Solution to [latex]4x-3=5[/latex]

[latex]x=2[/latex]

  1. [latex]3left(x+2right)-12=5left(x+1right)[/latex]
  2. [latex]12-5left(x+3right)=2x-5[/latex]
Show Solution

[latex]x=frac{2}{7}[/latex]

  1. [latex]frac{1}{2}-frac{1}{3}x=frac{4}{3}[/latex]
  2. [latex]frac{x}{3}-frac{3}{4}=frac{2x+3}{12}[/latex]
Show Solution

[latex]x=6[/latex]

  1. [latex]frac{2}{3}x+frac{1}{2}=frac{31}{6}[/latex]
  2. [latex]3left(2x-1right)+x=5x+3[/latex]
Show Solution

[latex]x=3[/latex]

  1. [latex]frac{2x}{3}-frac{3}{4}=frac{x}{6}+frac{21}{4}[/latex]
  2. [latex]frac{x+2}{4}-frac{x-1}{3}=2[/latex]
Show Solution

[latex]x=-14[/latex]

For the following exercises, solve each rational equation for[latex],x.,[/latex]State all x-values that are excluded from the solution set.

  1. [latex]frac{3}{x}-frac{1}{3}=frac{1}{6}[/latex]
  2. [latex]2-frac{3}{x+4}=frac{x+2}{x+4}[/latex]
Show Solution

[latex]xne -4;[/latex][latex]x=-3[/latex]

  1. [latex]frac{3}{x-2}=frac{1}{x-1}+frac{7}{left(x-1right)left(x-2right)}[/latex]
  2. [latex]frac{3x}{x-1}+2=frac{3}{x-1}[/latex]
Show Solution

[latex]xne 1;[/latex]when we solve this we get[latex],x=1,[/latex]which is excluded, therefore NO solution

  1. [latex]frac{5}{x+1}+frac{1}{x-3}=frac{-6}{{x}^{2}-2x-3}[/latex]
  2. [latex]frac{1}{x}=frac{1}{5}+frac{3}{2x}[/latex]
Show Solution

[latex]xne 0;[/latex][latex]x=frac{-5}{2}[/latex]

Glossary

conditional equation
an equation that is true for some values of the variable
identity equation
an equation that is true for all values of the variable
inconsistent equation
an equation producing a false result
linear equation
an algebraic equation in which each term is either a constant or the product of a constant and the first power of a variable
solution set
the set of all solutions to an equation
rational equation
an equation consisting of a fraction of polynomials

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2.1 Linear Equations in One Variable Copyright © by Cynthia Singleton; Ginny Bradley; Karen Perilloux; and Prakash Ghimire is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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