Explanation:
Let’s solve each linear equation one by one. The goal is to find the value of $ x $ that makes the equation true.
We’ll use these steps for each:
1. Expand any parentheses (distribute).
2. Combine like terms.
3. Move all terms with $ x $ to one side and constants to the other.
4. Solve for $ x $ by dividing.
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1. $ 5x + 2(2x - 4) = 55 $
- Expand: $ 5x + 4x - 8 = 55 $
- Combine: $ 9x - 8 = 55 $
- Add 8: $ 9x = 63 $
- Divide: $ x = 7 $
✔ Check: $ 5(7) + 2(2·7 - 4) = 35 + 2(14 - 4) = 35 + 20 = 55 $ ✔️
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2. $ 7x + 4(x - 9) = 41 $
- Expand: $ 7x + 4x - 36 = 41 $
- Combine: $ 11x - 36 = 41 $
- Add 36: $ 11x = 77 $
- Divide: $ x = 7 $
✔ Check: $ 7·7 + 4(7 - 9) = 49 + 4(-2) = 49 - 8 = 41 $ ✔️
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3. $ 3x + 2(x - 6) = 33 $
- Expand: $ 3x + 2x - 12 = 33 $
- Combine: $ 5x - 12 = 33 $
- Add 12: $ 5x = 45 $
- Divide: $ x = 9 $
✔ Check: $ 3·9 + 2(9 - 6) = 27 + 2·3 = 27 + 6 = 33 $ ✔️
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4. $ 4x + 3(3x + 7) = 86 $
- Expand: $ 4x + 9x + 21 = 86 $
- Combine: $ 13x + 21 = 86 $
- Subtract 21: $ 13x = 65 $
- Divide: $ x = 5 $
✔ Check: $ 4·5 + 3(3·5 + 7) = 20 + 3(15 + 7) = 20 + 3·22 = 20 + 66 = 86 $ ✔️
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5. $ 2x + 5(4x - 7) = 75 $
- Expand: $ 2x + 20x - 35 = 75 $
- Combine: $ 22x - 35 = 75 $
- Add 35: $ 22x = 110 $
- Divide: $ x = 5 $
✔ Check: $ 2·5 + 5(4·5 - 7) = 10 + 5(20 - 7) = 10 + 5·13 = 10 + 65 = 75 $ ✔️
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6. $ 97 = 7x + 2(x + 8) $
- Expand right side: $ 7x + 2x + 16 = 9x + 16 $
- So: $ 97 = 9x + 16 $
- Subtract 16: $ 81 = 9x $
- Divide: $ x = 9 $
✔ Check: $ 7·9 + 2(9 + 8) = 63 + 2·17 = 63 + 34 = 97 $ ✔️
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7. $ 6x + 5(2x - 7) = 45 $
- Expand: $ 6x + 10x - 35 = 45 $
- Combine: $ 16x - 35 = 45 $
- Add 35: $ 16x = 80 $
- Divide: $ x = 5 $
✔ Check: $ 6·5 + 5(2·5 - 7) = 30 + 5(10 - 7) = 30 + 5·3 = 30 + 15 = 45 $ ✔️
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8. $ 63 = 6x + 3(3x - 4) $
- Expand: $ 6x + 9x - 12 = 15x - 12 $
- So: $ 63 = 15x - 12 $
- Add 12: $ 75 = 15x $
- Divide: $ x = 5 $
✔ Check: $ 6·5 + 3(3·5 - 4) = 30 + 3(15 - 4) = 30 + 3·11 = 30 + 33 = 63 $ ✔️
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9. $ 3x + 4(x + 6) = 59 $
- Expand: $ 3x + 4x + 24 = 59 $
- Combine: $ 7x + 24 = 59 $
- Subtract 24: $ 7x = 35 $
- Divide: $ x = 5 $
✔ Check: $ 3·5 + 4(5 + 6) = 15 + 4·11 = 15 + 44 = 59 $ ✔️
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10. $ 82 = 2x + 2(3x + 5) $
- Expand: $ 2x + 6x + 10 = 8x + 10 $
- So: $ 82 = 8x + 10 $
- Subtract 10: $ 72 = 8x $
- Divide: $ x = 9 $
✔ Check: $ 2·9 + 2(3·9 + 5) = 18 + 2(27 + 5) = 18 + 2·32 = 18 + 64 = 82 $ ✔️
All solutions verified.
Final Answer:
x = 7, 7, 9, 5, 5, 9, 5, 5, 5, 9
Parent Tip: Review the logic above to help your child master the concept of solving linear equations worksheet with answers.
