Graph Linear Equations in Two Variables

Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate system is also called the xy-plane or the “coordinate plane.”

The rectangular coordinate system is formed by two intersecting number lines, one horizontal and one vertical. The horizontal number line is called the x-axis. The vertical number line is called the y-axis. These axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. See .

In the rectangular coordinate system, every point is represented by an ordered pair. The first number in the ordered pair is the x-coordinate of the point, and the second number is the y-coordinate of the point. The phrase “ordered pair” means that the order is important.

What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is \((0,0).\) The point \((0,0)\) has a special name. It is called the origin.

We use the coordinates to locate a point on the xy-plane. Let’s plot the point \((1,3)\) as an example. First, locate 1 on the x-axis and lightly sketch a vertical line through \(x=1.\) Then, locate 3 on the y-axis and sketch a horizontal line through \(y=3.\) Now, find the point where these two lines meet—that is the point with coordinates \((1,3).\) See .

Notice that the vertical line through \(x=1\) and the horizontal line through \(y=3\) are not part of the graph. We just used them to help us locate the point \((1,3).\)

When one of the coordinate is zero, the point lies on one of the axes. In the point \((0,4)\) is on the y-axis and the point \((-2,0)\) is on the x-axis.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

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.

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. But the arithmetic is easier if we make “good” choices for the values of x. This way we will avoid fractional answers, which are hard to graph precisely.

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 \(x=-3.\) This equation has only one variable, x. The equation says that x is always equal to\(-3,\) so its value does not depend on y. No matter what is the value of y, the value of x is always \(-3.\)

So to make a table of values, write \(-3\) in for all the x-values. Then choose any values for y. Since x does not depend on y, you can choose any numbers you like. But to fit the points on our coordinate graph, we’ll use 1, 2, and 3 for the y-coordinates. See .

Plot the points from the table and connect them with a straight line. Notice that we have graphed a vertical line.

What if the equation has y but no x? Let’s graph the equation \(y=4.\) 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’s in and then choose any values for x. We’ll 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.

What is the difference between the equations \(y=4x\) and \(y=4?\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

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.

First, notice where each of these lines crosses the x-axis. See .

Now, let’s look at the points where these lines cross the y-axis.

Do you see a pattern?

For each line, the y-coordinate of the point where the line crosses the x-axis is zero. The point where the line crosses the x-axis has the form \((a,0)\) and is called the x-intercept of the line. The x-intercept occurs when y is zero.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

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.

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

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

• Points on the Axes
  • Points with a y-coordinate equal to 0 are on the x-axis, and have coordinates \((a,0).\)
  • Points with an x-coordinate equal to \(0\) are on the y-axis, and have coordinates \((0,b).\)
• Quadrant \[\begin{array}{llll}\text{Quadrant I} & \ \text{Quadrant II} & \ \text{Quadrant III} & \ \text{Quadrant IV} \\ (x,y) & \ (x,y) & \ (x,y) & \ (x,y) \\ (+,+) & \ (-,+) & \ (-,-) & \ (+,-)\end{array}\]
• Graph of a Linear Equation: The graph of a linear equation \(Ax+By=C\) is a straight line.
  Every point on the line is a solution of the equation.
  Every solution of this equation is a point on this line.
• How to graph a linear equation by plotting points.
  1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
  2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
  3. Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.
• x-intercept and y-intercept of a Line
  • The x-intercept is the point \((a,0)\) where the line crosses the x-axis.
  • The y-intercept is the point \((0,b)\) where the line crosses the y-axis.
• Find the x- and y-intercepts from the Equation of a Line
  • Use the equation of the line. To find:
    the x-intercept of the line, let \(y=0\) and solve for x.
    the y-intercept of the line, let \(x=0\) and solve for y.
• How to graph a linear equation using the intercepts.
  1. Find the x- and y-intercepts of the line.
     Let \(y=0\) and solve for x.
     Let \(x=0\) and solve for y.
  2. Find a third solution to the equation.
  3. Plot the three points and check that they line up.
  4. Draw the line

Plot Points in a Rectangular Coordinate System

In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.

In the following exercises, for each ordered pair, decide

ⓐ is the ordered pair a solution to the equation? ⓑ is the point on the line?

Graph a Linear Equation by Plotting Points

In the following exercises, graph by plotting points.

Graph Vertical and Horizontal lines

In the following exercises, graph each equation.

In the following exercises, graph each pair of equations in the same rectangular coordinate system.

Find x- and y-Intercepts

In the following exercises, find the x- and y-intercepts on each graph.

In the following exercises, find the intercepts for each equation.

Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

Mixed Practice

In the following exercises, graph each equation.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

In the previous section, we found several solutions to the equation \(3x+2y=6\). They are listed in . So, the ordered pairs \((0,3)\), \((2,0)\), and \((1,\frac{3}{2})\) are some solutions to the equation \(3x+2y=6\). We can plot these solutions in the rectangular coordinate system as shown in .

Notice how the points line up perfectly? We connect the points with a line to get the graph of the equation \(3x+2y=6\). See . Notice the arrows on the ends of each side of the line. These arrows indicate the line continues.

Every point on the line is a solution of the equation. Also, every solution of this equation is a point on this line. Points not on the line are not solutions.

Notice that the point whose coordinates are \((-2,6)\) is on the line shown in . If you substitute \(x=-2\) and \(y=6\) into the equation, you find that it is a solution to the equation.

So the point \((-2,6)\) is a solution to the equation \(3x+2y=6\). (The phrase “the point whose coordinates are \((-2,6)\)” is often shortened to “the point \((-2,6)\).”)

So \((4,1)\) is not a solution to the equation \(3x+2y=6\). Therefore, the point \((4,1)\) is not on the line. See . This is an example of the saying, “A picture is worth a thousand words.” The line shows you all the solutions to the equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation \(3x+2y=6\).

There are several methods that can be used to graph a linear equation. The method we used to graph \(3x+2y=6\) is called plotting points, or the Point–Plotting Method.

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

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 part (a) and part (b) in .

Let’s do another example. This time, we’ll show the last two steps all on one grid.

When an equation includes a fraction as the coefficient of \(x\), we can still substitute any numbers for \(x\). But the math is easier if we make ‘good’ choices for the values of \(x\). This way we will avoid fraction answers, which are hard to graph precisely.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Can we graph an equation with only one variable? Just \(x\) and no \(y\), or just \(y\) without an \(x\)? How will we make a table of values to get the points to plot?

Let’s consider the equation \(x=-3\). This equation has only one variable, \(x\). The equation says that \(x\) is always equal to \(-3\), so its value does not depend on \(y\). No matter what \(y\) is, the value of \(x\) is always \(-3\).

So to make a table of values, write \(-3\) in for all the \(x\) values. Then choose any values for \(y\). Since \(x\) does not depend on \(y\), you can choose any numbers you like. But to fit the points on our coordinate graph, we’ll use 1, 2, and 3 for the y-coordinates. See .

Plot the points from and connect them with a straight line. Notice in that we have graphed a vertical line.

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

The graph is a horizontal line passing through the y-axis at 4. See .

Condensed — the full section is in OpenStax Elementary Algebra 2e.

• Graph a Linear Equation by Plotting Points
  1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
  2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work!
  3. Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.

Recognize the Relationship Between the Solutions of an Equation and its Graph

In the following exercises, for each ordered pair, decide:

ⓐ Is the ordered pair a solution to the equation? ⓑ Is the point on the line?

Graph a Linear Equation by Plotting Points

In the following exercises, graph by plotting points.

Graph Vertical and Horizontal Lines

In the following exercises, graph each equation.

In the following exercises, graph each pair of equations in the same rectangular coordinate system.

In the following exercises, graph each equation.

Condensed — the full section is in OpenStax Elementary Algebra 2e.