Point-Slope Equation of a Line Worksheets - Free Printable

Let's solve each problem step by step using the point-slope form of a line.

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🔷 Point-Slope Form:

The point-slope form of a line is:

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

Where:
- $ m $ is the slope,
- $ (x_1, y_1) $ is a point on the line.

We'll plug in the given values and simplify to get the equation of the line. Some problems may ask for standard or slope-intercept form, but since it says "write each equation", we’ll write it in point-slope form unless specified otherwise. However, often it's expected to simplify to slope-intercept form ($ y = mx + b $).

Let’s go through each one.

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1) $ m = 6 $ and $ (4, 12) $

Using point-slope:
$$
y - 12 = 6(x - 4)
$$

Simplify:
$$
y - 12 = 6x - 24 \\
y = 6x - 12
$$

✔ Answer: $ \boxed{y = 6x - 12} $

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2) $ m = \frac{1}{4} $ and $ (-3, -9) $

$$
y + 9 = \frac{1}{4}(x + 3)
$$

Simplify:
$$
y + 9 = \frac{1}{4}x + \frac{3}{4} \\
y = \frac{1}{4}x + \frac{3}{4} - 9 \\
y = \frac{1}{4}x + \frac{3}{4} - \frac{36}{4} \\
y = \frac{1}{4}x - \frac{33}{4}
$$

✔ Answer: $ \boxed{y = \frac{1}{4}x - \frac{33}{4}} $

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3) $ m = -7 $ and $ (-9, 5) $

$$
y - 5 = -7(x + 9)
$$

Simplify:
$$
y - 5 = -7x - 63 \\
y = -7x - 58
$$

✔ Answer: $ \boxed{y = -7x - 58} $

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4) $ m = 1 $ and $ (6, -10) $

$$
y + 10 = 1(x - 6) \\
y + 10 = x - 6 \\
y = x - 16
$$

✔ Answer: $ \boxed{y = x - 16} $

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5) $ m = -2 $ and $ (-1, -4) $

$$
y + 4 = -2(x + 1) \\
y + 4 = -2x - 2 \\
y = -2x - 6
$$

✔ Answer: $ \boxed{y = -2x - 6} $

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6) $ m = -\frac{3}{5} $ and $ (2, -3) $

$$
y + 3 = -\frac{3}{5}(x - 2)
$$

Simplify:
$$
y + 3 = -\frac{3}{5}x + \frac{6}{5} \\
y = -\frac{3}{5}x + \frac{6}{5} - 3 \\
y = -\frac{3}{5}x + \frac{6}{5} - \frac{15}{5} \\
y = -\frac{3}{5}x - \frac{9}{5}
$$

✔ Answer: $ \boxed{y = -\frac{3}{5}x - \frac{9}{5}} $

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7) $ m = -\frac{5}{8} $ and $ (-8, -7) $

$$
y + 7 = -\frac{5}{8}(x + 8)
$$

Simplify:
$$
y + 7 = -\frac{5}{8}x - 5 \\
y = -\frac{5}{8}x - 5 - 7 \\
y = -\frac{5}{8}x - 12
$$

✔ Answer: $ \boxed{y = -\frac{5}{8}x - 12} $

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8) $ m = 8 $ and $ (-5, 4) $

$$
y - 4 = 8(x + 5) \\
y - 4 = 8x + 40 \\
y = 8x + 44
$$

✔ Answer: $ \boxed{y = 8x + 44} $

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9) Slope $ \frac{1}{6} $, passes through $ (-9, 3) $

$$
y - 3 = \frac{1}{6}(x + 9)
$$

Simplify:
$$
y - 3 = \frac{1}{6}x + \frac{9}{6} \\
y - 3 = \frac{1}{6}x + \frac{3}{2} \\
y = \frac{1}{6}x + \frac{3}{2} + 3 \\
y = \frac{1}{6}x + \frac{3}{2} + \frac{6}{2} \\
y = \frac{1}{6}x + \frac{9}{2}
$$

✔ Answer: $ \boxed{y = \frac{1}{6}x + \frac{9}{2}} $

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✔ Final Answers Summary:

1) $ y = 6x - 12 $
2) $ y = \frac{1}{4}x - \frac{33}{4} $
3) $ y = -7x - 58 $
4) $ y = x - 16 $
5) $ y = -2x - 6 $
6) $ y = -\frac{3}{5}x - \frac{9}{5} $
7) $ y = -\frac{5}{8}x - 12 $
8) $ y = 8x + 44 $
9) $ y = \frac{1}{6}x + \frac{9}{2} $

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Let me know if you'd like them in standard form (Ax + By + C = 0) instead!