📝 Multiplication Property of Equality (14 MCQs)

📖 Explanation: The Multiplication Property of Equality states that if you multiply both sides of an equation by the same nonzero number, the sides remain equal. To isolate $x$, multiply both sides by 5: $5 \cdot \frac{x}{5} = -3 \cdot 5$, which gives $x = -15$. Option B is a common sign error, while C and D come from incorrectly multiplying or dividing.

📖 Explanation: Both A and C yield $x = 12$. For A, multiply both sides by 4: $x = 12$. For C, multiply both sides by 3: $x = 12$. Option B gives $x = \frac{3}{4}$. This question tests whether you understand that different equations can have the same solution and that the multiplication property works in both directions.

📖 Explanation: Start with $\frac{3}{4} = 12x$. To solve for $x$, multiply both sides by the reciprocal of 12, which is $\frac{1}{12}$: $x = \frac{3}{4} \cdot \frac{1}{12} = \frac{3}{48} = \frac{1}{16}$ cup per cookie. For 30 cookies, multiply by 30: $30 \cdot \frac{1}{16} = \frac{30}{16} = \frac{15}{8}$ cups. Option B is per-cookie amount for 10 cookies, C uses 12 as multiplier, D is per-cookie amount.

📖 Explanation: The student correctly identified that division is needed, but they divided by positive 4 instead of negative 4. Since the coefficient is -4, you must divide both sides by -4 to isolate $x$: $x = \frac{20}{-4} = -5$. Option B would give $x = -5$ if they had divided by -4, but they didn't. Option A and D are procedural errors not shown.

📖 Explanation: The equation $3x = 12$ asks: for what x-value does the line $y = 3x$ have a y-value of 12? Solving gives $x = 4$, which is the x-coordinate of the point where the horizontal line $y = 12$ intersects the graph. Option B reverses the variables, C confuses slope with solution, and D is the x-intercept (where y=0).

📖 Explanation: To isolate $x$, you must multiply by the reciprocal of $\frac{2}{3}$, which is $\frac{3}{2}$, because $\frac{2}{3} \cdot \frac{3}{2} = 1$. This gives $x = 8 \cdot \frac{3}{2} = 12$. Option A would leave $\frac{4}{9}x$, C is division by reciprocal which is same as multiplying by $\frac{2}{3}$, and D is not a multiplicative operation.

📖 Explanation: First solve for $x$: multiply both sides by -7 to get $x = -63$. The question asks for $-x$, so $-(-63) = 63$. Option A is the value of $x$, not $-x$. This tests if you can solve and then apply an additional operation, a common two-step reasoning task.

📖 Explanation: Solve $5x = 20$ gives $x = 4$. Solve $\frac{x}{5} = 20$ gives $x = 100$. Since 100 > 4, option C is correct. This question requires solving both equations and comparing the results, integrating multiplication and division properties in one problem.

📖 Explanation: The Multiplication Property of Equality says you can multiply both sides by the same nonzero number. Here, multiplying by 4 (the reciprocal of 0.25) isolates $x$. While the Multiplicative Inverse Property tells us that $0.25 \times 4 = 1$, the action of doing it to both sides is justified by the Multiplication Property of Equality. Option B is division, C is a property of numbers, D is order of operations.

📖 Explanation: Substitute $d = 240$ into $d = 60t$ to get $60t = 240$. Divide both sides by 60 (or multiply by $\frac{1}{60}$): $t = 4$ hours. Option B incorrectly multiplies 240 by 60, C uses division incorrectly, and D adds 60 to 240. This models real-world distance-rate-time relationships.

📖 Explanation: The student has a conceptual misunderstanding. For $\frac{x}{3} = 5$, multiply by 3: $x = 15$. For $3x = 5$, divide by 3: $x = \frac{5}{3}$. These are different operations (multiplication vs. division) and yield different solutions. This question tests the ability to recognize and explain the error in reasoning.

📖 Explanation: The line $y = -2x$ crosses the horizontal line $y = 10$ when $-2x = 10$. Divide both sides by -2 (or multiply by $-\frac{1}{2}$): $x = -5$. Option B has sign error, C uses wrong coefficient sign, D has wrong constant sign. This connects graphical interpretation with algebraic solution.

📖 Explanation: First solve $\frac{3x}{4} = 9$. Multiply both sides by 4: $3x = 36$, then divide by 3: $x = 12$. The question asks for $\frac{x}{4}$, which is $\frac{12}{4} = 3$. Option A is $x$, C is the original RHS, D is the denominator. This requires solving and then substituting into a different expression, a higher-order reasoning task.

📖 Explanation: To solve $\frac{x}{-5} = 15$, you must isolate $x$. Since $x$ is divided by -5, you multiply by -5. Multiplying by 5 would give $\frac{5x}{-5} = 75$ or $-x = 75$, which is not a direct isolation and introduces an extra step. Option C is equivalent to multiplying by -5 (since dividing by $\frac{1}{-5}$ is multiplying by -5), and D is the reciprocal of -5, so it’s also valid. This tests understanding of equivalent operations.