Graphing Linear Equations

Graphing a linear equation means drawing its straight line on a coordinate plane. There are three main methods to graph a linear equation: using the slope-intercept form, finding the intercepts, or making a table of values.

Method 1: Slope-Intercept Form

The slope-intercept form of a linear equation is written as: $y = mx + b$

• $m$ is the slope (rise over run).
• $b$ is the $y$-intercept (where the line crosses the $y$-axis).

Example: Graph $y = 2x - 1$

1. Find the $y$-intercept: Here, $b = -1$. Plot the point $(0, -1)$ on the $y$-axis.
2. Use the slope to find a second point: The slope is $m = 2$. Think of this as a fraction: $\frac{2}{1}$. From $(0, -1)$, go up 2 units and right 1 unit to land on $(1, 1)$. Plot this point.
3. Draw the line: Connect the points $(0, -1)$ and $(1, 1)$ with a straight line and extend it in both directions.

Method 2: Using Intercepts

This method works great for equations written in standard form ($Ax + By = C$). You only need to find where the line crosses the $x$-axis and the $y$-axis.

• $x$-intercept: Set $y = 0$ and solve for $x$.
• $y$-intercept: Set $x = 0$ and solve for $y$.

Example: Graph $3x + 2y = 12$

1. Find the $x$-intercept: $3x + 2(0) = 12$ $3x = 12 \implies x = 4$ Plot the point $(4, 0)$.
2. Find the $y$-intercept: $3(0) + 2y = 12$ $2y = 12 \implies y = 6$ Plot the point $(0, 6)$.
3. Draw the line: Connect $(4, 0)$ and $(0, 6)$ with a straight line.

Method 3: Making a Table of Values

If you are ever unsure, you can always pick random $x$-values, plug them into the equation to find the corresponding $y$-values, and plot the resulting coordinate pairs.

For example, to graph $y = -x + 3$:

• If $x = 0$, $y = -(0) + 3 = 3 \implies (0, 3)$
• If $x = 1$, $y = -(1) + 3 = 2 \implies (1, 2)$
• If $x = 2$, $y = -(2) + 3 = 1 \implies (2, 1)$

Plot these three points and draw a line through them.