Here are the step-by-step solutions for each problem on the worksheet.
Problem 1 (Top Left)
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Equation: $6 - 5y = 4(y - 3)$
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Step 1: Distribute the 4 on the right side: $6 - 5y = 4y - 12$
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Step 2: Add $5y$ to both sides: $6 = 9y - 12$
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Step 3: Add 12 to both sides: $18 = 9y$
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Step 4: Divide by 9: $y = 2$
Problem 2 (Top Right)
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Equation: $\frac{1}{3}x - \frac{1}{2}x = x + 3$
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Step 1: Find a common denominator (6) for the left side: $\frac{2}{6}x - \frac{3}{6}x = x + 3$
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Step 2: Combine terms on the left: $-\frac{1}{6}x = x + 3$
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Step 3: Subtract $x$ from both sides ($-\frac{1}{6}x - \frac{6}{6}x$): $-\frac{7}{6}x = 3$
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Step 4: Multiply by $-\frac{6}{7}$ to isolate $x$: $x = -\frac{18}{7}$
Problem 3 (Middle Left)
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Equation: $4(x - y) = 5$
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Step 1: Distribute the 4: $4x - 4y = 5$
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Step 2: Subtract $4x$ from both sides: $-4y = 5 - 4x$
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Step 3: Divide by -4: $y = \frac{5 - 4x}{-4}$ or $y = x - \frac{5}{4}$
Problem 4 (Middle Right)
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Equation: $y = 4(x - 6y) + 2$
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Step 1: Distribute the 4: $y = 4x - 24y + 2$
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Step 2: Add $24y$ to both sides: $25y = 4x + 2$
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Step 3: Divide by 25: $y = \frac{4x + 2}{25}$
Problem 5 (Bottom Left)
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Equation: $3(2y - 7) = z$
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Step 1: Distribute the 3: $6y - 21 = z$
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Step 2: Add 21 to both sides: $6y = z + 21$
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Step 3: Divide by 6: $y = \frac{z + 21}{6}$
Problem 6 (Bottom Right)
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Equation: $3 - y = x + 10$
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Step 1: Subtract 3 from both sides: $-y = x + 7$
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Step 2: Multiply by -1 to make $y$ positive: $y = -x - 7$
Final Answer:
1. $y = 2$
2. $x = -\frac{18}{7}$
3. $y = x - \frac{5}{4}$
4. $y = \frac{4x + 2}{25}$
5. $y = \frac{z + 21}{6}$
6. $y = -x - 7$
Parent Tip: Review the logic above to help your child master the concept of hard algebra y mx b worksheet.
