📝 Division and Multiplication Properties of Equality (17 MCQs)

📖 Explanation: The coefficient of $x$ is -5. To isolate $x$, we must perform the inverse operation, which is division by -5. The Division Property of Equality states that dividing both sides of an equation by the same non-zero number keeps the equation balanced. Option A is incorrect because dividing by 5 would leave $-x = 9$, not solving for $x$. Options C and D involve multiplication, which would not undo the multiplication by -5.

📖 Explanation: The student multiplied both sides of the equation by the same number (7) to maintain equality. This is a direct application of the Multiplication Property of Equality, which states that if $a = b$, then $a \cdot c = b \cdot c$. The Division Property involves dividing, the Additive Inverse involves addition, and the Commutative Property changes order, not balance.

📖 Explanation: Set up a proportion: $\frac{3/4}{12} = \frac{x}{20}$. Cross-multiply: $20 \cdot (3/4) = 12x \Rightarrow 15 = 12x \Rightarrow x = 1.25$. This requires using the Multiplication Property to eliminate the denominator. Option A (1 cup) underestimates, C (1.5) and D (2) overestimate due to incorrect scaling.

📖 Explanation: The first step $4x = 12$ is correct after subtracting 6. However, to isolate $x$, the student should divide both sides by 4, getting $x = 3$. Instead, they multiplied 12 by 4 to get 48, incorrectly applying the Multiplication Property when the Division Property was needed. This is a classic misconception of reversing operations.

📖 Explanation: Solving $3x = 15$ using the Division Property gives $x = 5$. For the equation $y = 3x$, when $x = 5$, $y = 15$, so the point is (5, 15). Option B reverses coordinates; C uses $x$ as $y$; D misinterprets the slope. This connects algebraic solution to graphical representation.

📖 Explanation: (Revised) In solving $-2x = 14$, a student divides by -2 and gets $x = -7$. Another student multiplies by $-\frac{1}{2}$ and gets $x = -7$. Which statement is true? Both are correct because dividing by -2 is equivalent to multiplying by its reciprocal $-\frac{1}{2}$. Option A says only division is correct, which is false. So A is the error.

📖 Explanation: To solve, multiply both sides by the reciprocal of $\frac{3}{5}$, which is $\frac{5}{3}$: $x = 9 \cdot \frac{5}{3} = 15$. Option B is $9 \div 3 \times 5$ misapplied; C is $9 \times 3$; D is $9 \times \frac{3}{5}$. This tests the concept that multiplying by the reciprocal is equivalent to division.

📖 Explanation: Multiply both sides by 2.5 using the Multiplication Property: $d = 60 \times 2.5 = 150$. Option B assumes $60 \times 2$; C assumes $60 \times 1.5$ misreading 2.5 as 1.5; D uses $60 \times 2.5$ but adds extra. This models a real-world proportional relationship.

📖 Explanation: Dividing by 0.2 is equivalent to multiplying by 5 because $0.2 = \frac{1}{5}$. Both methods correctly isolate $x$ and yield the same answer. Option A or B alone is incomplete; this tests the understanding of equivalent operations and decimal-fraction conversion.

📖 Explanation: First, multiply both sides by 3 (Multiplication Property) to get $2x - 4 = 24$. Then add 4 (Addition Property) to get $2x = 28$. Finally divide by 2 (Division Property) to get $x = 14$. Option B reverses addition/division; C ignores the order of operations; D applies division incorrectly. This requires ordering inverse operations.

📖 Explanation: The student divided both sides of the equation by 4 to isolate $s$. This is a direct application of the Division Property of Equality, which states that dividing both sides by the same non-zero number keeps the equation balanced. Multiplication would be used if the coefficient were a fraction; substitution and symmetric are not relevant here.

📖 Explanation: The equation is $\frac{x}{-3} = 7$. To isolate $x$, multiply both sides by -3 (the denominator) – this is the Multiplication Property. $x = 7 \times (-3) = -21$. So the student is correct. Option B is wrong because division would complicate; C is nonsense; D is a common error in sign. This tests the reciprocal relationship between multiplication and division.

📖 Explanation: The correct operation is to divide both sides by 8, giving $x = 3$. If you only divide the left side, the equation becomes $x = 24$, but that's not balanced. Wait: If you divide left side by 8, you get $x = 24$? No, $8x / 8 = x$, right side remains 24, so $x = 24$. Option A. But let's check: The student divided only the left side: $8x \div 8 = x$, right side stays 24, so $x = 24$. Option A. But D says $8x = 3$ which would be dividing right only. Let's adjust options. Actually, the correct answer is A. I'll set options: A) $x = 24$, B) $x = 3$, C) $x = 192$, D) $8x = 3$. The error is division on one side only, which violates the property. Answer A.

📖 Explanation: The equation $y = kx$ gives $18 = k \cdot 6$. To find $k$, divide both sides by 6 using the Division Property: $k = 3$. Option B is the reciprocal; C is the x-coordinate; D is the y-coordinate. This connects algebraic solution to slope interpretation from a graph.

📖 Explanation: The student multiplied both sides of the entire equation by 2: $2(0.5x + 3) = 2(7)$ gives $x + 6 = 14$. This is a valid application of the Multiplication Property of Equality because it's applied to both sides. Then subtracting 6 is valid. Option B is wrong because order can vary as long as properties are applied correctly. This tests the flexibility of solving multi-step equations.

📖 Explanation: Multiply both sides by the reciprocal $\frac{3}{2}$: $x = \frac{5}{6} \cdot \frac{3}{2} = \frac{15}{12} = \frac{5}{4}$. Option B is $\frac{5}{6} \cdot \frac{2}{3}$; C is the reciprocal of the answer; D is $\frac{6}{5} \cdot \frac{3}{2}$. This requires fraction multiplication and simplification, testing deeper understanding of reciprocals and equivalent forms.

📖 Explanation: The check involves substituting the found value back into the original equation. This is justified by the Substitution Property of Equality, which allows replacing a variable with its equal value. The Division Property was used to solve, but the verification uses substitution. Options A and B are operations; D is about equality of a number to itself. This distinguishes between solving and checking.