Lesson 3.1: Graph Linear Equations in Two Variables

New Concept

A linear equation in two variables, such as $Ax+By=C$, represents a straight line where every point is a solution. This lesson covers plotting points and using intercepts to accurately graph these lines on the coordinate plane.

What’s next

Next, you'll practice graphing by plotting points and using intercepts through a series of interactive examples and challenge problems.

Property

An ordered pair, $(x, y)$, gives the coordinates of a point in a rectangular coordinate system. The first number is the $x$-coordinate. The second number is the $y$-coordinate.

The point $(0, 0)$ is called the origin. It is the point where the $x$-axis and $y$-axis intersect.

Points with a $y$-coordinate equal to $0$ are on the $x$-axis, and have coordinates $(a, 0)$.

Property

An equation of the form $Ax + By = C$, where $A$ and $B$ are not both zero, is called a linear equation in two variables.

A linear equation is in standard form when it is written $Ax + By = C$.

An ordered pair $(x, y)$ is a solution of the linear equation $Ax + By = C$, if the equation is a true statement when the $x$- and $y$-values of the ordered pair are substituted into the equation.

Property

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

Examples

• To graph $y = x + 3$, find three solution points. If $x=0, y=3$, giving $(0, 3)$. If $x=1, y=4$, giving $(1, 4)$. If $x=-1, y=2$, giving $(-1, 2)$. Plot these points and draw the line through them.
• To graph $y = 2x - 1$, find three points. If $x=0, y=-1$, so $(0, -1)$. If $x=2, y=3$, so $(2, 3)$. If $x=-1, y=-3$, so $(-1, -3)$. Plot these and connect them.
• To graph $y = \frac{1}{3}x + 2$, choose multiples of 3 for $x$ to avoid fractions. If $x=0, y=2$, so $(0, 2)$. If $x=3, y=3$, so $(3, 3)$. If $x=-3, y=1$, so $(-3, 1)$. Plot and connect.

Explanation

To draw a line, you only need a few of its points. Find three $(x, y)$ pairs that solve the equation, plot them on the coordinate plane, and connect them with a straight line. Using three points helps you catch any calculation mistakes.

Property

A vertical line is the graph of an equation of the form $x = a$. The line passes through the $x$-axis at $(a, 0)$.

A horizontal line is the graph of an equation of the form $y = b$. The line passes through the $y$-axis at $(0, b)$.

Examples

• The graph of the equation $x = 4$ is a vertical line where every point has an x-coordinate of 4. Examples of points include $(4, 0)$, $(4, 2)$, and $(4, -5)$.
• The graph of the equation $y = -1$ is a horizontal line where every point has a y-coordinate of -1. Examples of points include $(0, -1)$, $(3, -1)$, and $(-2, -1)$.
• The equation $x = 0$ represents the $y$-axis, while the equation $y = 0$ represents the $x$-axis.

Property

The $x$-intercept is the point $(a, 0)$ where the line crosses the $x$-axis. The $x$-intercept occurs when $y$ is zero.

The $y$-intercept is the point $(0, b)$ where the line crosses the $y$-axis. The $y$-intercept occurs when $x$ is zero.

To find the $x$-intercept of the line, let $y = 0$ and solve for $x$.
To find the $y$-intercept of the line, let $x = 0$ and solve for $y$.

Property

Graph a linear equation using the intercepts.
Step 1. Find the $x$- and $y$-intercepts of the line. Let $y = 0$ and solve for $x$. Let $x = 0$ and solve for $y$.
Step 2. Find a third solution to the equation.
Step 3. Plot the three points and check that they line up.
Step 4. Draw the line.

Examples

• Graph $2x + 3y = 6$. The $x$-intercept (let $y=0$) is $(3, 0)$. The $y$-intercept (let $x=0$) is $(0, 2)$. A third point (let $x=-3$) is $(-3, 4)$. Plot these three points and draw the line.
• Graph $x - 2y = 4$. The $x$-intercept is $(4, 0)$ and the $y$-intercept is $(0, -2)$. For a third point, let $x=2$, then $2 - 2y = 4$, so $-2y=2$ and $y=-1$. The point is $(2, -1)$. Plot the three points and connect.
• Graph $y = 3x$. The $x$-intercept and $y$-intercept are both $(0, 0)$. We need two more points. If $x=1$, $y=3$, giving $(1, 3)$. If $x=-1$, $y=-3$, giving $(-1, -3)$. Plot $(0,0), (1,3), (-1,-3)$ and draw the line.

Explanation

This method is a graphing shortcut. Find the two points where the line hits the axes (the intercepts), plot them, find one more point as a backup check, and then connect them. It's often the fastest way to graph a line.