Linear Equations

What is a linear equation?

A linear equation is an equation whose graph is a straight line. The most useful form is slope-intercept form:

$y = mx + b$

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

Solving for x

To solve a linear equation, your job is to isolate $x$ on one side. Whatever you do to one side, do to the other.

Example. Solve $3x + 7 = 22$.

1. Subtract 7 from both sides: $3x = 15$
2. Divide both sides by 3: $x = 5$

Example with x on both sides. Solve $5x - 4 = 2x + 11$.

1. Subtract $2x$ from both sides: $3x - 4 = 11$
2. Add 4: $3x = 15$
3. Divide by 3: $x = 5$

Finding slope from two points

Given two points $(x_1, y_1)$ and $(x_2, y_2)$, slope is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Example. Slope through $(2, 3)$ and $(6, 11)$:

$m = \frac{11 - 3}{6 - 2} = \frac{8}{4} = 2$

Writing the equation of a line

If you know the slope $m$ and one point $(x_1, y_1)$, use point-slope form:

$y - y_1 = m(x - x_1)$

Then rearrange to $y = mx + b$ if you want slope-intercept form.

Key takeaways

• Linear equation → straight line graph.
• Slope $m$ = rise / run.
• Slope-intercept: $y = mx + b$.
• To solve: do the same thing to both sides until $x$ is alone.