📝 Addition Property of Equality (13 MCQs)

📖 Explanation: The student's equation $x = 3.8 - 7.2$ shows they moved 7.2 to the right side without changing its sign, which is a common 'transposing' error. The correct step is to add 7.2 to both sides: $x - 7.2 + 7.2 = 3.8 + 7.2$, giving $x = 11.0$. This is a conceptual misunderstanding of the Addition Property, which requires adding the same number to both sides to maintain equality.

📖 Explanation: To isolate $y$, you must undo the addition of $\frac{2}{3}$. The inverse operation is subtraction, so you subtract $\frac{2}{3}$ from both sides: $y = -\frac{1}{4} - \frac{2}{3} = -\frac{3}{12} - \frac{8}{12} = -\frac{11}{12}$. Option B is a common error of using the wrong inverse operation, and C/D show incorrect fraction addition.

📖 Explanation: The situation describes an increase in fuel, so addition is correct: original $g$ plus added 4.5 equals new total 11.3. To solve, subtract 4.5 from both sides: $g = 11.3 - 4.5 = 6.8$. Option D incorrectly adds the 4.5 to the final reading, which would double-count the added fuel. This is a direct application of the Addition Property in a real-world context.

📖 Explanation: The student's reasoning focuses on the sign of the left side, which is irrelevant. The goal is to isolate $x$, so you must undo the addition of 9 on the right side. The inverse operation is subtraction of 9 from both sides: $-5 - 9 = x + 9 - 9$, giving $x = -14$. Option A is a common distractor for those who confuse the sign of the result with the operation needed.

📖 Explanation: Moving right on a number line means adding a positive value. The equation is $p + 4 = 2$. To solve, subtract 4 from both sides (which is the Addition Property with a negative number): $p = 2 - 4 = -2$. The justification is the Addition Property of Equality (adding -4 to both sides). Option C incorrectly labels the property as 'Subtraction' when it's fundamentally addition of a negative.

📖 Explanation: Student A applies the Addition Property directly by adding 5.6 to both sides: $3.2 + 5.6 = m - 5.6 + 5.6$, giving $m = 8.8$. Student B subtracts 3.2 from both sides: $3.2 - 3.2 = m - 5.6 - 3.2$, which simplifies to $0 = m - 8.8$, then adding 8.8 to both sides yields $m = 8.8$. So both methods are mathematically valid if fully executed. Option A gives the wrong second solution. This is a conceptual comparison question.

📖 Explanation: Step 1: The equation is $\frac{x}{3} - 5 = 2$. To isolate the fraction, we add 5 to both sides: $\frac{x}{3} = 7$. This is the Addition Property of Equality. The step is correct. Option B is wrong because it's addition, not subtraction. Option C misunderstands which constant to target. This is a direct recall of the property's application in a multi-step context.

📖 Explanation: This question tests the understanding that the Addition Property can be applied in any valid order. The first student uses it first (adding -7), the second student uses it after dividing. Both paths are algebraically sound and lead to $x = 3$. Option A is correct. The other options represent common misconceptions about a fixed 'order of operations' for solving equations, which does not exist as long as equality is maintained.

📖 Explanation: On a number line, the equation $4 = 2 + t$ means that starting at 2, moving $t$ units lands at 4. The distance and direction are +2, so $t = 2$. Algebraically, subtract 2 from both sides: $4 - 2 = t$, so $t = 2$. Option B incorrectly interprets the direction. Option C mistakes the sign. This graph-based question requires visualizing the property as a translation on the number line.

📖 Explanation: The student correctly applies the Addition Property: add 8 to both sides of $-5 = c - 8$ to get $-5 + 8 = c$, which simplifies to $3 = c$. Checking: $3 - 8 = -5$, which is true. The student's reasoning about 'negative' is a misconception, but their operation was correct. Option D is a common error of adding a negative instead of the positive inverse of -8.

📖 Explanation: The equation is $a - 2.3 = -1.7$. To isolate $a$, you must add 2.3 to both sides: $a = -1.7 + 2.3 = 0.6$. The student's answer $-4.0$ indicates they subtracted 2.3 ($-1.7 - 2.3 = -4.0$), which is a classic error of using the wrong inverse operation. Option B describes correct operation but wrong sign handling, which is also possible but less likely given the exact value -4.0.

📖 Explanation: Given $x + p = q$, solving gives $x = q - p$. If $p$ and $q$ are negative, the sign of $x$ depends on their relative values. For example, $p = -3, q = -5$ gives $x = -2$ (negative), but $p = -5, q = -3$ gives $x = 2$ (positive). However, if $p = q$, then $x = 0$ regardless of their values. Option D is always true. This is an abstract reasoning question that tests the conceptual understanding of the property without numeric values.

📖 Explanation: Equation (I): $2x - 5 = 11$. Adding 5 to both sides (Addition Property) yields $2x - 5 + 5 = 11 + 5$, simplifying to $2x = 16$, which is equation (II). So the first student is correct about the transformation. The second student is correct that to solve for $x$, you must then divide by 2, but that's a different property. Option C captures that both statements are true in context. This question tests the distinction between transformation and solution steps.