Let’s solve each equation one by one. We’ll use inverse operations — that means we undo what’s being done to the variable, step by step, until we find its value.
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Problem 1: 12 × (11 + y) = 264
Step 1: Divide both sides by 12 to get rid of the multiplication.
→ (11 + y) = 264 ÷ 12
→ 11 + y = 22
Step 2: Subtract 11 from both sides to isolate y.
→ y = 22 - 11
→ y = 11
✔ Check: 12 × (11 + 11) = 12 × 22 = 264 → Correct!
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Problem 2: 10 × (9 + y) = 180
Step 1: Divide both sides by 10.
→ (9 + y) = 180 ÷ 10
→ 9 + y = 18
Step 2: Subtract 9 from both sides.
→ y = 18 - 9
→ y = 9
✔ Check: 10 × (9 + 9) = 10 × 18 = 180 → Correct!
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Problem 3: 5 × y - 12 = -7
Step 1: Add 12 to both sides to cancel out the -12.
→ 5y = -7 + 12
→ 5y = 5
Step 2: Divide both sides by 5.
→ y = 5 ÷ 5
→ y = 1
✔ Check: 5 × 1 - 12 = 5 - 12 = -7 → Correct!
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Problem 4: 8² + y² = 113
First, calculate 8²:
→ 8² = 64
So now:
→ 64 + y² = 113
Step 1: Subtract 64 from both sides.
→ y² = 113 - 64
→ y² = 49
Step 2: Take the square root of both sides.
→ y = √49
→ y = 7 or y = -7
But since this is pre-algebra and often they expect the positive answer unless told otherwise, we’ll go with
y = 7.
✔ Check: 8² + 7² = 64 + 49 = 113 → Correct!
*(Note: If negative answers are allowed, then y = -7 also works. But for now, we’ll list the positive one as main answer.)*
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Problem 5: 4 × y + 1 = 9
Step 1: Subtract 1 from both sides.
→ 4y = 9 - 1
→ 4y = 8
Step 2: Divide both sides by 4.
→ y = 8 ÷ 4
→ y = 2
✔ Check: 4 × 2 + 1 = 8 + 1 = 9 → Correct!
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Final Answer:
1. y = 11
2. y = 9
3. y = 1
4. y = 7
5. y = 2
Parent Tip: Review the logic above to help your child master the concept of solving equations algebra worksheet.
