Here are the step-by-step solutions for each equation on the worksheet.
1) $8(4a - 4) - 1 = 14 - 8a$
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Step 1: Distribute the 8 into the parentheses: $32a - 32 - 1 = 14 - 8a$.
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Step 2: Combine like terms on the left side: $32a - 33 = 14 - 8a$.
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Step 3: Add $8a$ to both sides to get variables together: $40a - 33 = 14$.
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Step 4: Add 33 to both sides: $40a = 47$.
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Step 5: Divide by 40: $a = \frac{47}{40}$ (or $1.175$).
2) $4(-8x + 5) = -30x - 26$
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Step 1: Distribute the 4: $-32x + 20 = -30x - 26$.
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Step 2: Add $32x$ to both sides: $20 = 2x - 26$.
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Step 3: Add 26 to both sides: $46 = 2x$.
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Step 4: Divide by 2: $x = 23$.
3) $27 = 46 + 3x - x$
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Step 1: Combine the $x$ terms on the right side ($3x - x = 2x$): $27 = 46 + 2x$.
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Step 2: Subtract 46 from both sides: $-19 = 2x$.
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Step 3: Divide by 2: $x = -\frac{19}{2}$ (or $-9.5$).
4) $38 + 7k = 8(k + 4)$
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Step 1: Distribute the 8 on the right side: $38 + 7k = 8k + 32$.
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Step 2: Subtract $7k$ from both sides: $38 = k + 32$.
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Step 3: Subtract 32 from both sides: $6 = k$.
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Answer: $k = 6$.
5) $-2(4 - x) = 6(x + 4) + 4x$
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Step 1: Distribute on both sides. Left: $-8 + 2x$. Right: $6x + 24 + 4x$.
Equation becomes: $-8 + 2x = 10x + 24$.
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Step 2: Subtract $2x$ from both sides: $-8 = 8x + 24$.
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Step 3: Subtract 24 from both sides: $-32 = 8x$.
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Step 4: Divide by 8: $x = -4$.
6) $0.05z + 0.2 = 0.15z - 10.5$
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Step 1: Subtract $0.05z$ from both sides: $0.2 = 0.10z - 10.5$.
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Step 2: Add 10.5 to both sides: $10.7 = 0.10z$.
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Step 3: Divide by 0.10 (which is the same as multiplying by 10): $z = 107$.
7) $4(6 + 4x) = 0$
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Step 1: Distribute the 4: $24 + 16x = 0$.
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Step 2: Subtract 24 from both sides: $16x = -24$.
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Step 3: Divide by 16: $x = -\frac{24}{16}$.
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Step 4: Simplify the fraction (divide top and bottom by 8): $x = -\frac{3}{2}$ (or $-1.5$).
8) $\frac{x}{8} - \frac{x}{9} = 1$
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Step 1: Find a common denominator for 8 and 9, which is 72.
Rewrite fractions: $\frac{9x}{72} - \frac{8x}{72} = 1$.
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Step 2: Combine the numerators: $\frac{1x}{72} = 1$.
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Step 3: Multiply both sides by 72: $x = 72$.
Final Answer:
1) $a = \frac{47}{40}$ or $1.175$
2) $x = 23$
3) $x = -9.5$
4) $k = 6$
5) $x = -4$
6) $z = 107$
7) $x = -1.5$
8) $x = 72$
Parent Tip: Review the logic above to help your child master the concept of solving equations worksheets.
