Lesson 2: Introduction to Graphing Linear Equations

Property

The graph of a linear equation $Ax + By = C$ is a line.

• Every point on the line is a solution of the equation.
• Every solution of this equation is a point on this line.

Examples

• For the equation $y = 2x + 1$, the point $(2, 5)$ is a solution because $5 = 2(2) + 1$. Therefore, the point $(2, 5)$ is on the graph of the line.
• For the same equation, $y = 2x + 1$, the point $(3, 6)$ is not a solution because $6 \neq 2(3) + 1$. Therefore, the point $(3, 6)$ is not on the line.
• Every point on the graph of $x + y = 5$, such as $(1, 4)$ and $(5, 0)$, represents a pair of numbers that are solutions to the equation.

Property

To 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 like $(0, 3)$, $(1, 4)$, and $(-1, 2)$. Plot these points and draw a line through them.
• To graph $y = \frac{1}{2}x - 1$, choose multiples of 2 for $x$ to avoid fractions. Good choices would be $(0, -1)$, $(2, 0)$, and $(4, 1)$.
• For $3x + y = 4$, first rewrite it as $y = -3x + 4$. Then find points by choosing values for $x$, such as $(0, 4)$, $(1, 1)$, and $(2, -2)$.

Explanation

This method is like an algebraic connect-the-dots. Find at least three coordinate pairs that solve the equation, place them on the graph, and draw a straight line through them. Using three points helps you catch any calculation mistakes.

Property

The slope of the line between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

Examples

• Find the slope between $(2, 3)$ and $(7, 9)$. Using the formula: $m = \frac{9 - 3}{7 - 2} = \frac{6}{5}$.
• Find the slope between $(-1, 5)$ and $(3, -3)$. Using the formula: $m = \frac{-3 - 5}{3 - (-1)} = \frac{-8}{4} = -2$.
• Find the slope between $(-4, -2)$ and $(-6, 8)$. Using the formula: $m = \frac{8 - (-2)}{-6 - (-4)} = \frac{10}{-2} = -5$.

Explanation

The slope formula is a way to calculate rise over run without a graph. It finds the vertical distance between points $(y_2 - y_1)$ and divides it by the horizontal distance $(x_2 - x_1)$ to find the steepness.