Training Lines and Systems of Linear Equations Writing Equations of Lines
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Writing Equations of Lines

20 min Lines and Systems of Linear Equations

Writing Equations of Lines

Writing the equation of a line is a skill you will use repeatedly in algebra and beyond. Given enough information — a slope and a point, two points, or a graph — you can always determine the unique equation of the line.

This lesson covers the point-slope form y − y₁ = m(x − x₁), converting between forms, and writing equations of parallel and perpendicular lines. These techniques connect geometry and algebra in a powerful way.

Point-Slope Form

Form

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

Use when you know a point and the slope.

Example 1

Line through $(3, -2)$ with slope $4$.

$y + 2 = 4(x - 3) \;\Rightarrow\; y = 4x - 14$

Given Two Points

Example 2

Line through $(1, 5)$ and $(4, 11)$.

$m = \dfrac{11 - 5}{4 - 1} = 2$. Using $(1, 5)$: $y - 5 = 2(x - 1) \;\Rightarrow\; y = 2x + 3$.

Parallel and Perpendicular Lines

Rules
  • Parallel: Same slope ($m_1 = m_2$)
  • Perpendicular: Negative reciprocal slopes ($m_1 \cdot m_2 = -1$)
Example 3

Line parallel to $y = 3x - 1$ through $(2, 4)$.

$m = 3$. $y - 4 = 3(x - 2) \;\Rightarrow\; y = 3x - 2$.

Example 4

Line perpendicular to $y = \dfrac{2}{5}x + 1$ through $(4, -1)$.

$m = -\dfrac{5}{2}$. $y + 1 = -\dfrac{5}{2}(x - 4) \;\Rightarrow\; y = -\dfrac{5}{2}x + 9$.

Practice Problems

1. Line through $(0, 3)$, slope $-2$.
2. Line through $(2, 7)$ and $(5, 1)$.
3. Parallel to $y = -x + 5$ through $(3, 2)$.
4. Perpendicular to $y = 4x$ through $(8, 1)$.
5. Line through $(-1, -4)$ and $(3, 8)$.
6. Horizontal line through $(5, -2)$.
7. Vertical line through $(5, -2)$.
8. Parallel to $2x + 3y = 6$ through $(0, 0)$.
9. Perpendicular to $y = \dfrac{1}{3}x - 2$ through $(1, 4)$.
10. Line through $(4, 0)$ and $(0, -3)$.
11. Find the $x$-intercept of $y = 2x - 6$.
12. Write in standard form: $y = \dfrac{3}{4}x - 2$.
Show Answer Key

1. $y = -2x + 3$

2. $y = -2x + 11$

3. $y = -x + 5$

4. $y = -\dfrac{1}{4}x + 3$

5. $y = 3x - 1$

6. $y = -2$

7. $x = 5$

8. $y = -\dfrac{2}{3}x$

9. $y = -3x + 7$

10. $y = \dfrac{3}{4}x - 3$

11. $(3, 0)$

12. $3x - 4y = 8$

Slope and Slope-Intercept Form Solving Systems by Substitution