📝 Subtraction Property of Equality (15 MCQs)

📖 Explanation: The Subtraction Property of Equality states that if you subtract the same number from both sides of an equation, the two sides remain equal. In this step, 7 is subtracted from both sides to isolate $x$, so this property is directly applied. The other options involve different operations.

📖 Explanation: To solve $y + 3.2 = 8.7$, subtract $3.2$ from both sides: $y = 8.7 - 3.2 = 5.5$. This is a straightforward application of the subtraction property, requiring only basic decimal subtraction. The incorrect options come from adding or misinterpreting the sign.

📖 Explanation: The student's arithmetic gives $x = 25$ correctly, but the step violates the Subtraction Property of Equality because they did not subtract 15 from both sides. The equation becomes $x = 40$ on the right only if the left is changed, losing equality. Option C identifies the correct numerical result while pointing out the flawed reasoning, which is the essence of error analysis.

📖 Explanation: The perimeter is $2L + 2W = 48$. Since $L = W + 6$, substitute to get $2(W+6) + 2W = 48$, which simplifies to $4W + 12 = 48$. Subtract 12 from both sides: $4W = 36$, then divide by 4 to get $W = 9$. Option C correctly shows the simplified equation and result. This requires translating a word problem into an equation and applying the subtraction property within multi-step reasoning.

📖 Explanation: To solve $3x + 5 = 2x + 12$, you want to collect variable terms on one side. Subtracting $2x$ from both sides gives $x + 5 = 12$, which is a valid application of the subtraction property. This step reduces the equation to a simpler form, showing conceptual understanding of the property beyond simple one-step problems.

📖 Explanation: Substituting $x = -5$ into the original equation: $-5 + 8 = 3$, which is true. The solution is correct because subtracting 8 from both sides yields $x = 3 - 8 = -5$. Option A correctly confirms and verifies the solution, while the others contain common sign errors or misconceptions about the subtraction property.

📖 Explanation: Starting from $x + 12 = 5$, adding 12 to both sides gives $x + 24 = 17$. This is algebraically valid but does not isolate $x$, and it unnecessarily complicates the equation. The correct step is to subtract 12 from both sides to get $x = -7$. Option A correctly shows the resulting equation and identifies the error in strategy, making it a strong error-analysis question.

📖 Explanation: The intersection point satisfies both equations: $x + 2.5 = 10$. Solving gives $x = 7.5$. Graphically, the horizontal line $y = 10$ and the line $y = x + 2.5$ intersect at $(7.5, 10)$. This requires interpreting a graph and connecting it to the subtraction property, making it a graph-based HOTS question.

📖 Explanation: Applying the subtraction property: $x = b - a$. Since $a > b$, $b - a$ is negative. Therefore, $x$ must be negative. This question goes beyond simple computation and requires understanding the relationship between variables and the effect of subtraction, which tests conceptual grasp of the property.

📖 Explanation: The equation $m + 150 = 340$ represents starting mileage plus trip distance. Subtract 150 from both sides: $m = 190$. The starting mileage was 190 miles. This is a real-world application that requires setting up the equation correctly and interpreting the result, testing the student's ability to model and solve.

📖 Explanation: For $x - 5 = 2$, the correct operation is to add 5, not subtract, because the variable has 5 subtracted from it. The subtraction property would not apply directly; the addition property is needed. Options A, B, and D are correctly solved using subtraction. This question tests the student's ability to distinguish when the subtraction property is appropriately applied versus when another operation is required.

📖 Explanation: The solution $x = 22$ is correct, and the check verifies it. Even though the steps may not be shown, the application of the subtraction property and the verification are valid. Option A correctly identifies that the reasoning is sound because the check confirms the solution. This question evaluates the student's ability to assess the validity of a solution process, which is a key HOTS skill.

📖 Explanation: Both equations are identical: $x + 7 = 10$ and $y + 7 = 10$. Applying the subtraction property to both gives $x = 3$ and $y = 3$, so $x = y$. This question combines the subtraction property with the transitive property of equality, testing the student's ability to connect multiple concepts.

📖 Explanation: Shifting $y = x + 3$ down by 3 gives $y = x$. The original equation $x + 3 = 7$ becomes $x = 4$ after the shift, because the horizontal line $y = 7$ now intersects $y = x$ at $x = 4$. This question requires understanding transformations and connecting them to the subtraction property, making it a graph-based HOTS question.

📖 Explanation: The equation is $x + 9 = 4$. Applying the subtraction property: $x = 4 - 9 = -5$. The student incorrectly added 9 to 4, confusing the operation. This is a classic reversal error. The correct solution is negative, which may be counterintuitive. This question is challenging because it requires careful reasoning about inverse operations and handling negative results, fitting an Olympiad-style problem.