Writing Linear Equations Worksheet with graph, table, and slope-intercept problems.

Let's solve each problem step by step.

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1. Write the slope-intercept form of the equation of each given line

Slope-intercept form:
$$
y = mx + b
$$
where:
- $ m $ is the slope
- $ b $ is the y-intercept

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#### Problem 1: First graph (left)

We observe the line passes through:
- The y-intercept is at $ (0, 3) $ → so $ b = 3 $
- Another point on the line: $ (-1, 0) $

Now calculate slope $ m $:

$$
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 3}{-1 - 0} = \frac{-3}{-1} = 3
$$

So the equation is:
$$
\boxed{y = 3x + 3}
$$

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#### Problem 2: Second graph (right)

The line passes through:
- Y-intercept: $ (0, 2) $ → $ b = 2 $
- Another point: $ (1, 0) $

Slope:
$$
m = \frac{0 - 2}{1 - 0} = \frac{-2}{1} = -2
$$

Equation:
$$
\boxed{y = -2x + 2}
$$

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2. Write a linear equation for each table relating x and y

We’ll use the formula $ y = mx + b $. We need to find the slope and y-intercept from the data.

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#### Problem 3: Table

| x | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| y | 5 | 8 | 11 | 14 |

Find the slope between two points:

From $ (2, 5) $ to $ (4, 8) $:
$$
m = \frac{8 - 5}{4 - 2} = \frac{3}{2} = 1.5
$$

Check with another pair: $ (4, 8) $ to $ (6, 11) $:
$$
m = \frac{11 - 8}{6 - 4} = \frac{3}{2} = 1.5
$$

So slope $ m = \frac{3}{2} $

Now find $ b $ using one point, say $ (2, 5) $:
$$
5 = \frac{3}{2}(2) + b \Rightarrow 5 = 3 + b \Rightarrow b = 2
$$

Equation:
$$
\boxed{y = \frac{3}{2}x + 2}
$$

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#### Problem 4: Table

| x | -5 | -2 | 0 | 3 | 5 |
|---|----|----|---|---|---|
| y | -16 | -7 | -1 | 8 | 14 |

First, notice that when $ x = 0 $, $ y = -1 $ → so $ b = -1 $

Now compute slope using two points, say $ (-2, -7) $ and $ (0, -1) $:
$$
m = \frac{-1 - (-7)}{0 - (-2)} = \frac{6}{2} = 3
$$

Check with $ (3, 8) $:
$ y = 3(3) - 1 = 9 - 1 = 8 $ ✔
$ (5, 14): 3(5) - 1 = 15 - 1 = 14 $ ✔

So equation:
$$
\boxed{y = 3x - 1}
$$

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3. Write the slope-intercept form using slope and y-intercept

Use $ y = mx + b $

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#### Problem 5:
- Slope $ m = -\frac{1}{2} $
- y-intercept $ b = -2 $

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

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#### Problem 6:
- Slope $ m = 2 $
- y-intercept $ b = 4 $

Equation:
$$
\boxed{y = 2x + 4}
$$

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✔ Final Answers:

1. $ y = 3x + 3 $
2. $ y = -2x + 2 $
3. $ y = \frac{3}{2}x + 2 $
4. $ y = 3x - 1 $
5. $ y = -\frac{1}{2}x - 2 $
6. $ y = 2x + 4 $

Let me know if you'd like these graphed or explained further!