Here are the step-by-step solutions for each problem on the worksheet.
1. Solve: $4(3x + 7) = 64 + 4x$
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Step 1: Distribute the 4 into the parentheses on the left side.
$$12x + 28 = 64 + 4x$$
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Step 2: Subtract $4x$ from both sides to move the variables to one side.
$$8x + 28 = 64$$
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Step 3: Subtract 28 from both sides to isolate the term with $x$.
$$8x = 36$$
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Step 4: Divide by 8.
$$x = \frac{36}{8}$$
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Step 5: Simplify the fraction (divide top and bottom by 4).
$$x = \frac{9}{2} \text{ or } 4.5$$
2. Solve: $16 = -4(2x - 5)$
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Step 1: Distribute the -4 into the parentheses. Be careful with signs!
$$16 = -8x + 20$$
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Step 2: Subtract 20 from both sides.
$$-4 = -8x$$
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Step 3: Divide by -8.
$$x = \frac{-4}{-8}$$
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Step 4: Simplify. Two negatives make a positive.
$$x = \frac{1}{2} \text{ or } 0.5$$
3. Solve: $-8(6 + 5x) = 3x - 5$
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Step 1: Distribute the -8.
$$-48 - 40x = 3x - 5$$
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Step 2: Add $40x$ to both sides to make the $x$ term positive (optional, but easier).
$$-48 = 43x - 5$$
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Step 3: Add 5 to both sides.
$$-43 = 43x$$
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Step 4: Divide by 43.
$$x = -1$$
4. Solve: $33 = -3(1 + 4x)$
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Step 1: Distribute the -3.
$$33 = -3 - 12x$$
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Step 2: Add 3 to both sides.
$$36 = -12x$$
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Step 3: Divide by -12.
$$x = -3$$
5. Solve: $\frac{4x - 2}{5} = 0.4$
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Step 1: Multiply both sides by 5 to remove the denominator.
$$4x - 2 = 2$$ *(because $0.4 \times 5 = 2$)*
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Step 2: Add 2 to both sides.
$$4x = 4$$
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Step 3: Divide by 4.
$$x = 1$$
6. Solve: $\frac{2}{3}x - 6 = \frac{1}{3}x + 3$
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Step 1: Subtract $\frac{1}{3}x$ from both sides.
$$\frac{1}{3}x - 6 = 3$$
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Step 2: Add 6 to both sides.
$$\frac{1}{3}x = 9$$
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Step 3: Multiply by 3 to isolate $x$.
$$x = 27$$
7. Find the mistake and correct it: $\frac{1}{4}(12x - 16) = 2x$
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The Mistake: The student forgot to distribute the $\frac{1}{4}$ to the second number inside the parentheses ($-16$). They only multiplied $\frac{1}{4}$ by $12x$.
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Correction:
1. Distribute $\frac{1}{4}$ to both terms:
$$3x - 4 = 2x$$
2. Subtract $2x$ from both sides:
$$x - 4 = 0$$
3. Add 4 to both sides:
$$x = 4$$
8. Find the mistake and correct it: $-4 + 14x = -12 + 12x$
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The Mistake: Looking at the likely error in student work for this type of problem, students often subtract incorrectly or combine constants wrong. Let's solve it correctly to find the answer.
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Correction:
1. Subtract $12x$ from both sides:
$$-4 + 2x = -12$$
2. Add 4 to both sides:
$$2x = -8$$
3. Divide by 2:
$$x = -4$$
Final Answer:
1. x = 4.5
2. x = 0.5
3. x = -1
4. x = -3
5. x = 1
6. x = 27
7. Mistake: Did not distribute 1/4 to -16. Correct Answer: x = 4
8. Correct Answer: x = -4
Parent Tip: Review the logic above to help your child master the concept of equations with distributive property worksheet.
