Linear equation worksheet with problems and graphs for practice.

Let's solve each problem step by step. We'll use the point-slope form of a linear equation:

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

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

We can then simplify to slope-intercept form:
$$
y = mx + b
$$

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1. Line going through the point (2, 1) with a slope of 7

Given: $ m = 7 $, $ (x_1, y_1) = (2, 1) $

Use point-slope form:
$$
y - 1 = 7(x - 2)
$$
Simplify:
$$
y - 1 = 7x - 14 \\
y = 7x - 13
$$

✔ Answer: $ \boxed{y = 7x - 13} $

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2. Line going through the point (2, -4) with a slope of -1

Given: $ m = -1 $, $ (x_1, y_1) = (2, -4) $

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

✔ Answer: $ \boxed{y = -x - 2} $

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3. Line going through the point (3, 1) with a slope of $ \frac{1}{2} $

Given: $ m = \frac{1}{2} $, $ (x_1, y_1) = (3, 1) $

$$
y - 1 = \frac{1}{2}(x - 3) \\
y - 1 = \frac{1}{2}x - \frac{3}{2} \\
y = \frac{1}{2}x - \frac{3}{2} + 1 \\
y = \frac{1}{2}x - \frac{1}{2}
$$

✔ Answer: $ \boxed{y = \frac{1}{2}x - \frac{1}{2}} $

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4. Line going through the point (-1, 2) with a slope of $ \frac{2}{3} $

Given: $ m = \frac{2}{3} $, $ (x_1, y_1) = (-1, 2) $

$$
y - 2 = \frac{2}{3}(x + 1) \\
y - 2 = \frac{2}{3}x + \frac{2}{3} \\
y = \frac{2}{3}x + \frac{2}{3} + 2 \\
y = \frac{2}{3}x + \frac{8}{3}
$$

✔ Answer: $ \boxed{y = \frac{2}{3}x + \frac{8}{3}} $

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5. Line going through the points (-4, 3) and (7, -1)

First, find the slope:
$$
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 3}{7 - (-4)} = \frac{-4}{11}
$$

Now use point-slope with one point, say $ (-4, 3) $:
$$
y - 3 = -\frac{4}{11}(x + 4)
$$
Simplify:
$$
y - 3 = -\frac{4}{11}x - \frac{16}{11} \\
y = -\frac{4}{11}x - \frac{16}{11} + 3 \\
y = -\frac{4}{11}x + \frac{17}{11}
$$

✔ Answer: $ \boxed{y = -\frac{4}{11}x + \frac{17}{11}} $

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6. Line going through the points (-3, -3) and (-1, 4)

Find slope:
$$
m = \frac{4 - (-3)}{-1 - (-3)} = \frac{7}{2}
$$

Use point $ (-3, -3) $:
$$
y + 3 = \frac{7}{2}(x + 3) \\
y + 3 = \frac{7}{2}x + \frac{21}{2} \\
y = \frac{7}{2}x + \frac{21}{2} - 3 \\
y = \frac{7}{2}x + \frac{15}{2}
$$

✔ Answer: $ \boxed{y = \frac{7}{2}x + \frac{15}{2}} $

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7. Graph (Left graph)

From the graph:
- The line passes through $ (-3, -3) $ and $ (0, 0) $
- So it goes through the origin → $ y = mx $
- Slope: $ m = \frac{0 - (-3)}{0 - (-3)} = \frac{3}{3} = 1 $

So equation: $ y = x $

✔ Answer: $ \boxed{y = x} $

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8. Graph (Right graph)

From the graph:
- It appears to pass through $ (-5, -2) $ and $ (5, 0) $
- Let's verify slope:
$$
m = \frac{0 - (-2)}{5 - (-5)} = \frac{2}{10} = \frac{1}{5}
$$

Now use point $ (5, 0) $:
$$
y - 0 = \frac{1}{5}(x - 5) \\
y = \frac{1}{5}x - 1
$$

Check with other point: $ x = -5 $
$$
y = \frac{1}{5}(-5) - 1 = -1 - 1 = -2 \quad \text{✓ matches}
$$

✔ Answer: $ \boxed{y = \frac{1}{5}x - 1} $

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✔ Final Answers Summary:

1. $ y = 7x - 13 $
2. $ y = -x - 2 $
3. $ y = \frac{1}{2}x - \frac{1}{2} $
4. $ y = \frac{2}{3}x + \frac{8}{3} $
5. $ y = -\frac{4}{11}x + \frac{17}{11} $
6. $ y = \frac{7}{2}x + \frac{15}{2} $
7. $ y = x $
8. $ y = \frac{1}{5}x - 1 $

Let me know if you'd like these in standard form or graphed!