Lesson 2: Graphing Linear Equations

New Concept

A linear equation's graph is a straight line where every point on the line is a solution. This lesson shows you how to draw these lines by plotting points, including special cases for vertical and horizontal lines.

What’s next

Next, you’ll work through interactive examples of plotting points. Then, you'll tackle a series of practice cards to master graphing different types of lines.

Property

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.

Examples

• Is the point $(2, 7)$ a solution to the equation $y = 3x + 1$? Yes, because substituting the values gives $7 = 3(2) + 1$, which simplifies to $7 = 7$. The point is on the line.

• Is the point $(1, 3)$ a solution to the equation $y = 3x + 1$? No, because substituting the values gives $3 = 3(1) + 1$, which simplifies to $3 = 4$. This is false, so the point is not on the line.

Property

Step 1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
Step 2. Plot the points on 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 points. Extend the line to fill the grid and put arrows on both ends of the line.

Examples

• To graph $y = x + 2$, find three solution points. If $x=0, y=2$, so we have $(0, 2)$. If $x=1, y=3$, so we have $(1, 3)$. If $x=-1, y=1$, so we have $(-1, 1)$. Plot these and draw the line.

• To graph $x+y=5$, find three points. If $x=0, y=5 \rightarrow (0, 5)$. If $x=1, y=4 \rightarrow (1, 4)$. If $x=4, y=1 \rightarrow (4, 1)$. Plot these points and connect them with a line.

Property

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

Examples

• The graph of the equation $x = 3$ is a vertical line where every point has an $x$-coordinate of $3$. Examples include $(3, 0)$, $(3, 2)$, and $(3, -4)$.

• The graph of $x = -4$ is a vertical line passing through the $x$-axis at $-4$. All points on the line, such as $(-4, 1)$ and $(-4, 5)$, have an $x$-coordinate of $-4$.

Property

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

Examples

• The graph of the equation $y = 5$ is a horizontal line where every point has a $y$-coordinate of $5$. Examples include $(0, 5)$, $(2, 5)$, and $(-1, 5)$.

• The graph of $y = -2$ is a horizontal line passing through the $y$-axis at $-2$. All points on the line, such as $(1, -2)$ and $(4, -2)$, have a $y$-coordinate of $-2$.