Graph a Linear Equation by Plotting Points

There are several methods that can be used to graph a linear equation. The first method we will use is called plotting points, or the Point-Plotting Method. We find three points whose coordinates are solutions to the equation and then plot them in a rectangular coordinate system. By connecting these points in a line, we have the graph of the linear equation.

Example 3

How to Graph a Linear Equation by Plotting Points

Graph the equation [latex]y=2x+1[/latex] by plotting points.

Exercise 3

Graph the equation by plotting points: [latex]y=2x-3[/latex]. Check your solutions to ensure they are on the line in the graph below.

Solution

 

 

 

Graph the equation by plotting points: [latex]y=-2x+4[/latex]. Check your solutions to ensure they are on the line in the graph below. 

Solution

 

The steps to take when graphing a linear equation by plotting points are summarized here.

It is true that it only takes two points to determine a line, but it is a good habit to use three points. If you only plot two points and one of them is incorrect, you can still draw a line, but it will not represent the solutions to the equation. It will be the wrong line.

If you use three points and one is incorrect, the points will not line up. This tells you something is wrong and you need to check your work. Look at the difference between these illustrations.

When an equation includes a fraction as the coefficient of x, we can still substitute any numbers for x. The arithmetic is easier if we make “good” choices for the values of x. Zero is a good choice for the value of x, and multiples of the denominator are good choices, because they will eliminate the fraction. This will avoid fractional answers, which are hard to graph precisely.

Example 4

Graph the equation: [latex]y=\frac{1}{2}x+3[/latex].

Find three points that are solutions to the equation. Since this equation has the fraction [latex]\frac{1}{2}[/latex] as a coefficient of x, we will choose values of x carefully. We will use 0 as one choice and multiples of 2 for the other choices. Why are multiples of 2 a good choice for values of x? By choosing multiples of 2, the multiplication by [latex]\frac{1}{2}[/latex] simplifies to a whole number.

[latex]\begin{eqnarray} x &=& 0\\ y &=& \frac{1}{2}x+3\\ y &=& \frac{1}{2}({0})+3\\ y &=& 0+3\\ y &=& 3\\ \end{eqnarray}[/latex]

[latex]\begin{eqnarray} x &=& 2\\ y &=& \frac{1}{2}x+3\\ y &=& \frac{1}{2}({2})+3\\ y &=& 1+3\\ y &=& 4\\ \end{eqnarray}[/latex]

[latex]\begin{eqnarray} x &=& 4\\ y &=& \frac{1}{2}x+3\\ y &=& \frac{1}{2}({4})+3\\ y &=& 2+3\\ y &=& 5\\ \end{eqnarray}[/latex]

The points are shown in the figure below.

[latex]y=\frac{1}{2}x+3[/latex]
[latex]x[/latex]	[latex]y[/latex]	[latex](x,y)[/latex]
[latex]0[/latex]	[latex]3[/latex]	[latex](0,3)[/latex]
[latex]2[/latex]	[latex]4[/latex]	[latex](2,4)[/latex]
[latex]4[/latex]	[latex]5[/latex]	[latex](4,5)[/latex]

Plot the points, check that they line up, and draw the line.

 

Exercise 4

Graph the equation: [latex]y=\frac{1}{3}x-1[/latex].

Solution

 

 

Graph the equation: [latex]y=\frac{1}{4}x+2[/latex].

Solution

 

Graph Vertical and Horizontal Lines

Some linear equations have only one variable. They may have just x and no y or just y without an x. This changes how we make a table of values to get the points to plot.

Let’s consider the equation [latex]x=-3[/latex]. This equation has only one variable, x. The equation says that x is always equal to [latex]-3[/latex], so its value does not depend on y. Regardless of the value of y, the value of x is always [latex]-3[/latex].

To make a table of values, write [latex]-3[/latex] in for all the x-values, then choose any values for y. We will use 1, 2, and 3 for the y-coordinates.

Plot the points from the table and connect them with a straight line. Notice that we have graphed a vertical line passing through the x-axis at [latex]-3[/latex].

What if the equation has y, but no x? Let’s graph the equation [latex]y=4[/latex]. This time the y-value is a constant, so in this equation, y does not depend on x. Fill in 4 for all the y-values in the table, then choose any values for x. We will use 0, 2, and 4 for the x-coordinates.

In this figure, we have graphed a horizontal line passing through the y-axis at 4.

Example 5

Graph the following lines.

a) [latex]x=2[/latex]

The equation has only one variable, x, and x is always equal to 2. We create a table where x is always 2 and then put in any values for y. The graph is a vertical line passing through the x-axis at 2.

[latex]x=2[/latex]
x	y	[latex](x,y)[/latex]
2	1	[latex](2,1)[/latex]
2	2	[latex](2,2)[/latex]
2	3	[latex](2,3)[/latex]

 

b) [latex]y=-1[/latex]

Similarly, the equation [latex]y=-1[/latex] has only one variable, y. The value of y is constant. All the ordered pairs in the next table have the same y-coordinate. The graph is a horizontal line passing through the y-axis at [latex]-1[/latex].

[latex]y=-1[/latex]
x	y	[latex](x,y)[/latex]
[latex]0[/latex]	[latex]-1[/latex]	[latex](0,-1)[/latex]
[latex]3[/latex]	[latex]-1[/latex]	[latex](3,-1)[/latex]
[latex]-3[/latex]	[latex]-1[/latex]	[latex](-3,-1)[/latex]

 

Exercise 5

Graph the equations:

a) [latex]x=5[/latex]

b) [latex]y=-4[/latex]

Solution

a)

b)

 

 

Graph the equations:

a) [latex]x=-2[/latex]

b) [latex]y=3[/latex]

Solution

a)

b)

What is the difference between the equations [latex]y=4x[/latex] and [latex]y=4[/latex]?

The equation [latex]y=4x[/latex] has both x and y. The value of y depends on the value of x, so the y -coordinate changes according to the value of x. The equation [latex]y=4[/latex] has only one variable. The value of y is constant, it does not depend on the value of x, so the y-coordinate is always 4.

Notice, in the graph, the equation [latex]y=4x[/latex] gives a slanted line, while [latex]y=4[/latex] gives a horizontal line.

Example 6

Graph [latex]y=-3x[/latex] and [latex]y=-3[/latex] in the same rectangular coordinate system.

We notice that the first equation has the variable x, while the second does not. We make a table of points for each equation and then graph the lines. The two graphs are shown.

 

 

Exercise 6

Graph the equations in the same rectangular coordinate system: [latex]y=-4x[/latex] and [latex]y=-4[/latex].

Solution

 

 

Graph the equations in the same rectangular coordinate system: [latex]y=3[/latex] and [latex]y=3x[/latex]

Solution

 

Find x– and y-intercepts

Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.

At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the x-axis and the y-axis. These points are called the intercepts of a line.

Let’s look at the graphs of the lines.

Let’s look at the points where these lines cross the x-axis and the y-axis.

Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the x- and y-intercepts as two of your three points. Find the intercepts and then find a third point to ensure accuracy. Make sure the points line up—then draw the line. This method is often the quickest way to graph a line.

Example 9

How to Graph a Line Using the Intercepts

Graph [latex]-x+2y=6[/latex] using the intercepts.

Exercise 9

Graph using the intercepts: [latex]x-2y=4[/latex].

Solution

 

Graph using the intercepts: [latex]-x+3y=6[/latex].

Solution

 

The steps to graph a linear equation using the intercepts are summarized here:

Example 10

Graph [latex]4x-3y=12[/latex] using the intercepts.

Find the intercepts and a third point.

x-intercept, let y=0	y-intercept, let x=0	third point, let y=4
[latex]\begin{eqnarray} 4x-3y &=& 12\\ 4x-3({0}) &=& 12\\ 4x &=& 12\\ x &=& 3\\ \end{eqnarray}[/latex]	[latex]\begin{eqnarray} 4x-3y &=& 12\\ 4({0})-3y &=& 12\\ -3y &=& 12\\ y &=& -4\\ \end{eqnarray}[/latex]	[latex]\begin{eqnarray} 4x-3y &=& 12\\ 4x-3({4}) &=& 12\\ 4x-12 &=& 12\\ 4x &=& 24\\ x &=& 6\\ \end{eqnarray}[/latex]

We list the points in the table and show the graph.

[latex]4x-3y=12[/latex]
[latex]x[/latex]	[latex]y[/latex]	[latex](x,y)[/latex]
[latex]3[/latex]	[latex]0[/latex]	[latex](3,0)[/latex]
[latex]0[/latex]	[latex]-4[/latex]	[latex](0,-4)[/latex]
[latex]6[/latex]	[latex]4[/latex]	[latex](6,4)[/latex]

 

Exercise 10

Graph using the intercepts: [latex]5x-2y=10[/latex].

Solution

 

 

Graph using the intercepts: [latex]3x-4y=12[/latex].

Solution

 

When the line passes through the origin, the x-intercept and the y-intercept are the same point.

Example 11

Graph [latex]y=5x[/latex] using the intercepts.

x-intercept, let [latex]y=0[/latex].	y-intercept, let [latex]x=0[/latex].
[latex]\begin{eqnarray} y &=& 5x\\ {0} &=& 5x\\ 0 &=& x\\ \end{eqnarray}[/latex]	[latex]\begin{eqnarray} y &=& 5x\\ y &=& 5\cdot{0}\\ y &=& 0\\ \end{eqnarray}[/latex]
[latex](0,0)[/latex]	[latex](0,0)[/latex]

This line has only one intercept. It is the point [latex](0,0)[/latex].

To ensure accuracy, we need to plot three points. Since the x– and y-intercepts are the same point, we need two more points to graph the line.

Let [latex]x=1[/latex].	Let [latex]x=-1[/latex].
[latex]\begin{eqnarray} y &=& 5x\\ y &=& 5\cdot{1}\\ y &=& 5\\ \end{eqnarray}[/latex]	[latex]\begin{eqnarray} y &=& 5x\\ y &=& 5{-1}\\ y &=& -5\\ \end{eqnarray}[/latex]

The resulting three points are summarized in the table.

[latex]y=5x[/latex]
[latex]x[/latex]	[latex]y[/latex]	[latex](x,y)[/latex]
[latex]0[/latex]	[latex]0[/latex]	[latex](0,0)[/latex]
[latex]1[/latex]	[latex]5[/latex]	[latex](1,5)[/latex]
[latex]-1[/latex]	[latex]-5[/latex]	[latex](-1,-5)[/latex]

Plot the three points, check that they line up, and draw the line.