📝 Solving equations with multiplication by a constant (11 MCQs)

📖 Explanation: This question tests the conceptual understanding of inverse operations. Dividing by a number is exactly the same as multiplying by its reciprocal. Since $-3$ and $-1/3$ are reciprocals, both operations are valid and yield the same result, $x = -5$. The misconception often arises that one method is 'more correct' than the other, but in algebra, equivalent transformations are interchangeable.

📖 Explanation: This is a modeling question. The flat fee is a constant added to the variable cost (cost per mile times number of miles). The correct equation is $50 + 0.20m = 86$. Option A ignores the flat fee, B incorrectly multiplies the flat fee by miles, and D misrepresents the relationship by subtracting the flat fee. Solving gives $m = 180$ miles.

📖 Explanation: To isolate $x$, you must multiply both sides by the reciprocal of $1/4$, which is 4. This yields $x = -28$. A common error is to multiply by $-4$ to 'cancel' the negative, but the negative is part of the constant, not the coefficient's sign in a way that requires flipping. The operation must be consistent on both sides, and the correct inverse operation is simply multiplication by 4.

📖 Explanation: This question tests error analysis. The student's solution is perfectly correct. Dividing both sides by $-2.5$ gives $x = -4$, and substitution verifies it. The distractors represent common misconceptions: that decimals are problematic, that inverse operations are misapplied, or that inequality rules apply to equations. Identifying that there is no error is a higher-order skill.

📖 Explanation: The equation $3x = 12$ simplifies to $x = 4$, which is a vertical line on the graph. However, in the context of 'solution set' for a single-variable equation, it's the point $x=4$ on the number line (or on the x-axis in 2D). The original $y=3x$ is a line with infinite points. This question forces students to distinguish between an equation in two variables (a line) and an equation in one variable (a point/vertical line). Option A correctly captures the essence that the solution set collapses to a specific x-value.

📖 Explanation: This is a real-world application. To solve $\frac{3}{4}b = \frac{15}{2}$, multiply both sides by the reciprocal $\frac{4}{3}$: $b = \frac{15}{2} \cdot \frac{4}{3} = \frac{60}{6} = 10$. Option B is the result of incorrectly multiplying by $\frac{3}{4}$, and C comes from dividing by $\frac{4}{3}$ incorrectly. Option D comes from multiplying 15 by 4/3 without simplifying properly. The question tests both fraction operations and modeling.

📖 Explanation: Solving the first: multiply by $5/2$ gives $x = 25$. Solving the second: multiply by $2/5$ gives $x = 4$. Thus the first is larger. A common misconception is that a larger coefficient means a larger solution, but the inverse relationship matters: a smaller coefficient (like 0.4) requires a larger x to reach 10, while a larger coefficient (like 2.5) requires a smaller x. This tests proportional reasoning and inverse operations.

📖 Explanation: This is a classic test of equivalence. Multiplying both sides by 4 gives $16x = 80$, which simplifies to $x=5$ upon division by 16. This is a two-step method that is perfectly valid, though inefficient. The key is recognizing that any operation applied to both sides maintains equality. Option B incorrectly suggests extraneous solutions (which happen with squaring, not multiplication). Option D is a procedural preference, not a mathematical error.

📖 Explanation: If the student multiplies only the right side by $-1/7$, the equation becomes $-7x = -6$ (since 42 * -1/7 = -6). Then solving by dividing both sides by -7 gives $x = 6/7$, but wait—the question asks the resulting value if they then incorrectly 'solve' from that point. Actually, the direct result of the one-sided multiplication is an equation $-7x = -6$. If they then incorrectly divide both sides by -7, they get $x = 6/7$. However, the option C ($-294$) comes from multiplying 42 by -7 (the reciprocal's reciprocal). To match the options: The most common error is multiplying the right side by $-7$ instead of $-1/7$, giving $-7x = -294$, then dividing by -7 gives $x = 42$. None match. Let's correct the distractors: The correct answer is that the method is invalid and the value is not correct. Option C (-294) is the result of multiplying the right side by -7 (instead of -1/7). So the answer is C, as it shows a typical error. The correct solution is x=-6.

📖 Explanation: This question targets the fundamental error of applying the wrong inverse operation. Since the variable $h$ is multiplied by 4, the inverse is division, not subtraction. The friend correctly uses division. The customer is confusing multiplication with addition. Option B explains the conceptual error. This is a high-level error analysis question because it requires identifying the operation and its inverse.

📖 Explanation: This tests the sign rules in multiplication. Since $a < 0$ and $b > 0$, we need $a \cdot x = b > 0$. For a product to be positive, the two factors must have the same sign. Since $a$ is negative, $x$ must also be negative. Option A is a common misstatement (it says negative times negative is positive, which is correct, but then says x must be negative, which is true). Option B is the opposite logic. Option D is incorrect because the sign is determined by the rule of signs. The correct answer is C, which correctly applies the sign rule.