Linear Equations and Tables of Values (A) Worksheet | PDF ... - Free Printable

Let's solve each of the tables based on the given straight-line equations in the form $ y = mx + c $. We'll use the rule:

- The y-value when $ x = 0 $ is $ c $ (the y-intercept).
- For every increase of 1 in $ x $, $ y $ increases by $ m $ (the gradient).
- So we can either:
- Plug in each $ x $ into the equation to find $ y $, or
- Start at $ x = 0 $, then add/subtract $ m $ for each step.

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✔ Step-by-step solution:

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#### 1. $ y = 4x - 2 $

- When $ x = 0 $, $ y = -2 $ → this is $ c $
- $ m = 4 $ → so $ y $ increases by 4 each time

| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|----|----|----|---|---|---|---|---|
| y | | | | -2 | | | | |

Now work left and right from $ x = 0 $:

- $ x = -1 $: $ y = -2 - 4 = -6 $
- $ x = -2 $: $ y = -6 - 4 = -10 $
- $ x = -3 $: $ y = -10 - 4 = -14 $
- $ x = 1 $: $ y = -2 + 4 = 2 $
- $ x = 2 $: $ y = 2 + 4 = 6 $
- $ x = 3 $: $ y = 6 + 4 = 10 $
- $ x = 4 $: $ y = 10 + 4 = 14 $

✔ Table:

| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|----|----|----|---|---|---|---|---|
| y | -14 | -10 | -6 | -2 | 2 | 6 | 10 | 14 |

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#### 2. $ y = 3x + 2 $

- $ c = 2 $ → $ y = 2 $ when $ x = 0 $
- $ m = 3 $ → $ y $ increases by 3

| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|----|----|----|---|---|---|---|---|
| y | | | | 2 | | | | |

Work out:

- $ x = -1 $: $ 2 - 3 = -1 $
- $ x = -2 $: $ -1 - 3 = -4 $
- $ x = -3 $: $ -4 - 3 = -7 $
- $ x = 1 $: $ 2 + 3 = 5 $
- $ x = 2 $: $ 5 + 3 = 8 $
- $ x = 3 $: $ 8 + 3 = 11 $
- $ x = 4 $: $ 11 + 3 = 14 $

✔ Table:

| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|----|----|----|---|---|---|---|---|
| y | -7 | -4 | -1 | 2 | 5 | 8 | 11 | 14 |

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#### 3. $ y = 0.5x + 1 $

- $ c = 1 $ → $ y = 1 $ at $ x = 0 $
- $ m = 0.5 $ → $ y $ increases by 0.5 each time

| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|----|----|----|---|---|---|---|---|
| y | | | | 1 | | | | |

Work out:

- $ x = -1 $: $ 1 - 0.5 = 0.5 $
- $ x = -2 $: $ 0.5 - 0.5 = 0 $
- $ x = -3 $: $ 0 - 0.5 = -0.5 $
- $ x = 1 $: $ 1 + 0.5 = 1.5 $
- $ x = 2 $: $ 1.5 + 0.5 = 2 $
- $ x = 3 $: $ 2 + 0.5 = 2.5 $
- $ x = 4 $: $ 2.5 + 0.5 = 3 $

✔ Table:

| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|----|----|----|---|---|---|---|---|
| y | -0.5 | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |

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#### 4. $ y = -x + 3 $

- $ c = 3 $ → $ y = 3 $ at $ x = 0 $
- $ m = -1 $ → $ y $ decreases by 1 each time

| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|----|----|----|---|---|---|---|---|
| y | | | | 3 | | | | |

Work out:

- $ x = -1 $: $ 3 - (-1) = 3 + 1 = 4 $? Wait — better to use: $ y = -x + 3 $

So:

- $ x = -3 $: $ y = -(-3) + 3 = 3 + 3 = 6 $
- $ x = -2 $: $ y = -(-2) + 3 = 2 + 3 = 5 $
- $ x = -1 $: $ y = -(-1) + 3 = 1 + 3 = 4 $
- $ x = 0 $: $ y = 0 + 3 = 3 $
- $ x = 1 $: $ y = -1 + 3 = 2 $
- $ x = 2 $: $ y = -2 + 3 = 1 $
- $ x = 3 $: $ y = -3 + 3 = 0 $
- $ x = 4 $: $ y = -4 + 3 = -1 $

✔ Table:

| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|----|----|----|---|---|---|---|---|
| y | 6 | 5 | 4 | 3 | 2 | 1 | 0 | -1 |

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#### 5. $ y = -2x - 4 $

- $ c = -4 $ → $ y = -4 $ at $ x = 0 $
- $ m = -2 $ → $ y $ decreases by 2 each time

| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|----|----|----|---|---|---|---|---|
| y | | | | -4 | | | | |

Work out:

- $ x = -1 $: $ y = -2(-1) - 4 = 2 - 4 = -2 $
- $ x = -2 $: $ y = -2(-2) - 4 = 4 - 4 = 0 $
- $ x = -3 $: $ y = -2(-3) - 4 = 6 - 4 = 2 $
- $ x = 1 $: $ y = -2(1) - 4 = -2 - 4 = -6 $
- $ x = 2 $: $ y = -4 - 4 = -8 $
- $ x = 3 $: $ y = -6 - 4 = -10 $
- $ x = 4 $: $ y = -8 - 4 = -12 $

✔ Table:

| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|----|----|----|---|---|---|---|---|
| y | 2 | 0 | -2 | -4 | -6 | -8 | -10 | -12 |

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✔ Final Answer Tables:

#### 1. $ y = 4x - 2 $
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|----|----|----|---|---|---|---|---|
| y | -14 | -10 | -6 | -2 | 2 | 6 | 10 | 14 |

#### 2. $ y = 3x + 2 $
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|----|----|----|---|---|---|---|---|
| y | -7 | -4 | -1 | 2 | 5 | 8 | 11 | 14 |

#### 3. $ y = 0.5x + 1 $
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|----|----|----|---|---|---|---|---|
| y | -0.5 | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |

#### 4. $ y = -x + 3 $
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|----|----|----|---|---|---|---|---|
| y | 6 | 5 | 4 | 3 | 2 | 1 | 0 | -1 |

#### 5. $ y = -2x - 4 $
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|----|----|----|---|---|---|---|---|
| y | 2 | 0 | -2 | -4 | -6 | -8 | -10 | -12 |

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🔍 Explanation Summary:

- In a linear equation $ y = mx + c $, the value of $ y $ when $ x = 0 $ is always $ c $.
- The gradient $ m $ tells you how much $ y $ changes for each unit change in $ x $.
- If $ m $ is positive, $ y $ increases as $ x $ increases.
- If $ m $ is negative, $ y $ decreases.
- You can compute values directly using substitution, or build the sequence step-by-step from $ x = 0 $ using $ \Delta y = m $.

This method creates an arithmetic sequence in the $ y $-values, which is why it's useful for graphing and understanding linear patterns.

Let me know if you'd like a visual graph of any of these!