Writing Linear Equations

Slope-intercept form and point-slope form

Writing Linear Equations

Slope-Intercept Form

$y = mx + b$

$m$ = slope
$b$ = y-intercept

Use when: You know the slope and y-intercept

Point-Slope Form

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

Use when: You know the slope and one point $(x_1, y_1)$

Finding Equation from Two Points

Steps:

Find the slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Use point-slope form with either point
Simplify to slope-intercept form

Special Lines

Horizontal line: $y = k$ (slope = 0) Vertical line: $x = h$ (undefined slope)

Parallel and Perpendicular Lines

Parallel: Same slope

$y = 2x + 3$ and $y = 2x - 1$ are parallel

Perpendicular: Slopes are negative reciprocals

$y = 2x + 3$ and $y = -\frac{1}{2}x + 1$ are perpendicular

📚 Practice Problems

1Problem 1easy

❓ Question:

Write an equation for a line with slope 3 and y-intercept -2

💡 Show Solution

Use slope-intercept form: $y = mx + b$

Given: $m = 3$ and $b = -2$

Substitute: $y = 3x + (-2)$ $y = 3x - 2$

Answer: $y = 3x - 2$

2Problem 2medium

❓ Question:

Write an equation for the line passing through $(2, 5)$ with slope $-4$

💡 Show Solution

Use point-slope form: $y - y_1 = m(x - x_1)$

Given: $(x_1, y_1) = (2, 5)$ and $m = -4$

Substitute: $y - 5 = -4(x - 2)$

Expand: $y - 5 = -4x + 8$

Add 5 to both sides: $y = -4x + 13$

Answer: $y = -4x + 13$

3Problem 3hard

❓ Question:

Write an equation for the line passing through $(1, 3)$ and $(4, 9)$

💡 Show Solution

Step 1: Find the slope $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 3}{4 - 1} = \frac{6}{3} = 2$

Step 2: Use point-slope form with $(1, 3)$ $y - 3 = 2(x - 1)$

Step 3: Simplify to slope-intercept form $y - 3 = 2x - 2$ $y = 2x + 1$

Answer: $y = 2x + 1$

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