Solving with Variables on Both Sides worksheet - Free Printable

Let's solve each equation step by step and check the solutions. Then we'll tackle the word problem at the end.

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1. $ x + 16 = 9x $

Step 1: Subtract $ x $ from both sides:
$$
x + 16 - x = 9x - x \\
16 = 8x
$$

Step 2: Divide both sides by 8:
$$
x = \frac{16}{8} = 2
$$

✔ Check:
Left side: $ x + 16 = 2 + 16 = 18 $
Right side: $ 9x = 9(2) = 18 $
✔ Both sides equal → Solution is correct

Answer: $ x = 2 $

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2. $ 4y - 70 = 12y + 2 $

Step 1: Subtract $ 4y $ from both sides:
$$
4y - 70 - 4y = 12y + 2 - 4y \\
-70 = 8y + 2
$$

Step 2: Subtract 2 from both sides:
$$
-70 - 2 = 8y \\
-72 = 8y
$$

Step 3: Divide by 8:
$$
y = \frac{-72}{8} = -9
$$

✔ Check:
Left: $ 4(-9) - 70 = -36 - 70 = -106 $
Right: $ 12(-9) + 2 = -108 + 2 = -106 $
✔ Equal → Correct

Answer: $ y = -9 $

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3. $ 5(p + 6) = 8p $

Step 1: Distribute the 5:
$$
5p + 30 = 8p
$$

Step 2: Subtract $ 5p $ from both sides:
$$
30 = 3p
$$

Step 3: Divide by 3:
$$
p = 10
$$

✔ Check:
Left: $ 5(10 + 6) = 5(16) = 80 $
Right: $ 8(10) = 80 $
✔ Correct

Answer: $ p = 10 $

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4. $ 3(g - 7) = 2(10 + g) $

Step 1: Distribute both sides:
$$
3g - 21 = 20 + 2g
$$

Step 2: Subtract $ 2g $ from both sides:
$$
g - 21 = 20
$$

Step 3: Add 21 to both sides:
$$
g = 41
$$

✔ Check:
Left: $ 3(41 - 7) = 3(34) = 102 $
Right: $ 2(10 + 41) = 2(51) = 102 $
✔ Correct

Answer: $ g = 41 $

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5. $ 1.8 + 7n = 9.5 - 4n $

Step 1: Add $ 4n $ to both sides:
$$
1.8 + 11n = 9.5
$$

Step 2: Subtract 1.8 from both sides:
$$
11n = 9.5 - 1.8 = 7.7
$$

Step 3: Divide by 11:
$$
n = \frac{7.7}{11} = 0.7
$$

✔ Check:
Left: $ 1.8 + 7(0.7) = 1.8 + 4.9 = 6.7 $
Right: $ 9.5 - 4(0.7) = 9.5 - 2.8 = 6.7 $
✔ Correct

Answer: $ n = 0.7 $

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6. $ \frac{3}{7}w - 11 = -\frac{4}{7}w $

Step 1: Add $ \frac{4}{7}w $ to both sides:
$$
\frac{3}{7}w + \frac{4}{7}w - 11 = 0 \\
\frac{7}{7}w - 11 = 0 \\
w - 11 = 0
$$

Step 2: Add 11 to both sides:
$$
w = 11
$$

✔ Check:
Left: $ \frac{3}{7}(11) - 11 = \frac{33}{7} - 11 = \frac{33 - 77}{7} = \frac{-44}{7} $
Right: $ -\frac{4}{7}(11) = -\frac{44}{7} $
✔ Equal → Correct

Answer: $ w = 11 $

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7. Movie Club Word Problem

Let $ m $ = number of movies

- Club A: $100 membership + $10 per movie → Cost = $ 100 + 10m $
- Club B: No membership, $15 per movie → Cost = $ 15m $

Set costs equal:
$$
100 + 10m = 15m
$$

Step 1: Subtract $ 10m $ from both sides:
$$
100 = 5m
$$

Step 2: Divide by 5:
$$
m = 20
$$

✔ Check:
Club A: $ 100 + 10(20) = 100 + 200 = 300 $
Club B: $ 15(20) = 300 $
✔ Same cost → Correct

Answer: You need to buy 20 movies for the cost to be the same.

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✔ Final Answers:

1. $ x = 2 $
2. $ y = -9 $
3. $ p = 10 $
4. $ g = 41 $
5. $ n = 0.7 $
6. $ w = 11 $
7. $ m = 20 $ movies

All solutions checked and verified!