Worksheet for solving equations with variables on both sides of the equal sign, showing problems and handwritten solutions.

Let's solve each equation step by step. I'll go through all the problems on your worksheet, explaining each solution clearly.

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1) $ 2a - 8a = 12 - 8a $

Step 1: Combine like terms on the left side:
$$
(2a - 8a) = -6a
$$
So the equation becomes:
$$
-6a = 12 - 8a
$$

Step 2: Add $ 8a $ to both sides to get all variables on one side:
$$
-6a + 8a = 12
\Rightarrow 2a = 12
$$

Step 3: Divide both sides by 2:
$$
a = \frac{12}{2} = 6
$$

✔ Answer: $ a = 6 $
*(You already got this correct!)*

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2) $ r - 6 = 6 + 4r $

Step 1: Move all variable terms to one side and constants to the other.

Subtract $ 4r $ from both sides:
$$
r - 4r - 6 = 6
\Rightarrow -3r - 6 = 6
$$

Step 2: Add 6 to both sides:
$$
-3r = 6 + 6 = 12
$$

Step 3: Divide by -3:
$$
r = \frac{12}{-3} = -4
$$

✘ You wrote $ r = -3 $, but that’s incorrect.

✔ Correct Answer: $ r = -4 $

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3) $ 1 + 4x = -5 + 7x $

Step 1: Subtract $ 7x $ from both sides:
$$
1 + 4x - 7x = -5
\Rightarrow 1 - 3x = -5
$$

Step 2: Subtract 1 from both sides:
$$
-3x = -5 - 1 = -6
$$

Step 3: Divide by -3:
$$
x = \frac{-6}{-3} = 2
$$

✔ Answer: $ x = 2 $

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4) $ 5r - 2 = 6r + 5 $

Step 1: Subtract $ 6r $ from both sides:
$$
5r - 6r - 2 = 5
\Rightarrow -r - 2 = 5
$$

Step 2: Add 2 to both sides:
$$
-r = 5 + 2 = 7
$$

Step 3: Multiply both sides by -1:
$$
r = -7
$$

✔ Answer: $ r = -7 $

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5) $ 5x + 7 = -14 + 8x $

Step 1: Subtract $ 8x $ from both sides:
$$
5x - 8x + 7 = -14
\Rightarrow -3x + 7 = -14
$$

Step 2: Subtract 7 from both sides:
$$
-3x = -14 - 7 = -21
$$

Step 3: Divide by -3:
$$
x = \frac{-21}{-3} = 7
$$

✔ Answer: $ x = 7 $

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6) $ -2 + 5n = 4n + 2 $

Step 1: Subtract $ 4n $ from both sides:
$$
-2 + 5n - 4n = 2
\Rightarrow -2 + n = 2
$$

Step 2: Add 2 to both sides:
$$
n = 2 + 2 = 4
$$

✔ Answer: $ n = 4 $

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7) $ 4p - 3 = -9 + 7p $

Step 1: Subtract $ 7p $ from both sides:
$$
4p - 7p - 3 = -9
\Rightarrow -3p - 3 = -9
$$

Step 2: Add 3 to both sides:
$$
-3p = -9 + 3 = -6
$$

Step 3: Divide by -3:
$$
p = \frac{-6}{-3} = 2
$$

✔ Answer: $ p = 2 $

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8) $ -4 - 5k = -5k - 4 $

Step 1: Add $ 5k $ to both sides:
$$
-4 - 5k + 5k = -4
\Rightarrow -4 = -4
$$

This is always true — it’s an identity.

✔ Answer: Infinite solutions (all real numbers)
Because both sides are identical after simplification.

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1) | $ a = 6 $ |
| 2) | $ r = -4 $ ✘ (you had -3) |
| 3) | $ x = 2 $ |
| 4) | $ r = -7 $ |
| 5) | $ x = 7 $ |
| 6) | $ n = 4 $ |
| 7) | $ p = 2 $ |
| 8) | Infinite solutions |

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