Detailed educational worksheet on writing linear equations from a table, covering slope-intercept form, slope calculation, and y-intercept, with step-by-step instructions and examples.

Let's solve the problem step-by-step using the information provided in the image. The goal is to write linear equations from a table using the slope-intercept form:
$$
y = mx + b
$$
where:
- $ m $ is the slope
- $ b $ is the y-intercept

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🔹 Step 1: Understand the Steps

The worksheet gives two cases:

#### ✔ Case 1: If $ x = 0 $ is in the table
1. Find the slope from the table.
2. Identify the y-intercept (the $ y $ value when $ x = 0 $).
3. Substitute slope ($ m $) and y-int ($ b $) into $ y = mx + b $.

#### ✘ Case 2: If $ x = 0 $ is NOT in the table
1. Find the slope.
2. Substitute your $ x $ and $ y $ values into the equation.
3. Substitute the slope and the point into $ y = mx + b $.
4. Solve for $ b $.
5. Write the equation with $ m $ and $ b $.

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Now let’s go through each example and fill in the blanks.

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## 🟦 Example 1: Table with $ x = 0 $

| x | -1 | 0 | 1 | 2 |
|---|----|---|---|---|
| y | 0 | 2 | 4 | 6 |

Step 1: Find the slope $ m $

Use two points: say $ (-1, 0) $ and $ (0, 2) $

$$
m = \frac{\Delta y}{\Delta x} = \frac{2 - 0}{0 - (-1)} = \frac{2}{1} = 2
$$

✔ Slope $ m = 2 $

Step 2: Find y-intercept $ b $

When $ x = 0 $, $ y = 2 $ → So $ b = 2 $

Step 3: Write equation

$$
y = mx + b = 2x + 2
$$

✔ Final answer: $ y = 2x + 2 $

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## 🟦 Example 2: Table with $ x = 0 $

| x | -2 | 0 | 2 | 4 |
|---|----|---|---|---|
| y | 5 |10 |15 |20 |

Step 1: Find the slope $ m $

Use $ (0, 10) $ and $ (2, 15) $

$$
m = \frac{15 - 10}{2 - 0} = \frac{5}{2} = 2.5
$$

Or as fraction: $ \frac{5}{2} $

✔ Slope $ m = \frac{5}{2} $

Step 2: Y-intercept $ b $

When $ x = 0 $, $ y = 10 $ → $ b = 10 $

Step 3: Equation

$$
y = \frac{5}{2}x + 10
$$

✔ Final answer: $ y = \frac{5}{2}x + 10 $

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## 🟦 Example 3: Table without $ x = 0 $

| x | 3 | 6 | 12 | 15 |
|---|----|----|----|----|
| y | 20 | 16 | 8 | 4 |

We need to find the slope first.

Step 1: Find slope $ m $

Use two points: $ (3, 20) $ and $ (6, 16) $

$$
m = \frac{16 - 20}{6 - 3} = \frac{-4}{3} = -\frac{4}{3}
$$

✔ Slope $ m = -\frac{4}{3} $

Step 2: Pick a point to substitute

Let’s use $ (3, 20) $

So we plug into:
$$
y = mx + b
$$
$$
20 = -\frac{4}{3}(3) + b
$$
$$
20 = -4 + b
$$
$$
b = 24
$$

✔ $ b = 24 $

Step 3: Write equation

$$
y = -\frac{4}{3}x + 24
$$

✔ Final answer: $ y = -\frac{4}{3}x + 24 $

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✔ Summary of Filled-in Worksheet:

#### Example 1:
- Slope: $ m = \frac{\Delta y}{\Delta x} = \frac{2 - 0}{0 - (-1)} = \frac{2}{1} = 2 $
- Y-int: $ b = 2 $
- Equation: $ y = 2x + 2 $

#### Example 2:
- Slope: $ m = \frac{15 - 10}{2 - 0} = \frac{5}{2} $
- Y-int: $ b = 10 $
- Equation: $ y = \frac{5}{2}x + 10 $

#### Example 3:
- Slope: $ m = \frac{16 - 20}{6 - 3} = \frac{-4}{3} $
- Point: $ (3, 20) $
- Plug in:
$$
20 = -\frac{4}{3}(3) + b \Rightarrow 20 = -4 + b \Rightarrow b = 24
$$
- Equation: $ y = -\frac{4}{3}x + 24 $

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🔁 Final Answers:

#### Example 1:
$$
y = 2x + 2
$$

#### Example 2:
$$
y = \frac{5}{2}x + 10
$$

#### Example 3:
$$
y = -\frac{4}{3}x + 24
$$

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✔ Key Takeaways:

- Always calculate slope using $ m = \frac{\Delta y}{\Delta x} $
- If $ x = 0 $ is in the table, read $ b $ directly.
- If not, pick any point, plug into $ y = mx + b $, and solve for $ b $.
- Then write the final equation.

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