📝 Collect constant terms on the other side (14 MCQs)

📖 Explanation: The student violated the fundamental property of equality: any operation performed on one side must be done on the other. Subtracting 7 only from the left changes the equation's balance. The correct first step is $3x + 7 - 7 = 22 - 7$, yielding $3x = 15$. This misconception often arises from treating equations as one-sided simplifications rather than balanced scales.

📖 Explanation: Adding 9 first changes the left side to $5x$ and the right to $2x + 15$, collecting constants on the right. This is strategic because it creates a simpler constant (15) and allows you to then subtract $2x$ to get $3x = 15$. The order is flexible, but this approach minimizes fractions and negative numbers, making the solution less error-prone.

📖 Explanation: The original equation is $0.06P + 500 = 8300$. To solve for $P$, you must isolate the term with $P$ by subtracting the constant 500 from both sides, giving $0.06P = 7800$ (not 8800). Then divide by 0.06. Option A correctly shows the subtraction, and C is the final formula. This models a real-world commission problem where the constant represents a base fee.

📖 Explanation: The student effectively added 3 to both sides (to move -3) and subtracted $4x$ from both sides (to move the variable term) simultaneously. This is valid because the Addition Property of Equality allows adding any term to both sides. Doing both at once is a shortcut, but it's mathematically sound as long as signs are correctly changed. This shows a strong conceptual grasp of maintaining balance.

📖 Explanation: After distribution: $6x - 8 + 5 = 3x + 10$ simplifies to $6x - 3 = 3x + 10$. To collect constant terms on the right, add 3 to both sides: $6x = 3x + 13$. Then subtract $3x$: $3x = 13$. However, the question asks for the step after collecting constants on the right, which is $6x - 3x = 10 + 3$. Option D correctly shows the constant 3 moving to the right as +3.

📖 Explanation: The student's step is valid: adding 10 to both sides moves the constant from right to left, yielding $8x + 15 = 3x$. Then subtract $3x$: $5x = -15$, so $x = -3$. This is a correct alternative path, showing constants can be collected on either side. The misconception that constants 'must' go to one side is false; the goal is isolation. The error in option C is sign error in combining.

📖 Explanation: Intersection means $4x - 7 = 10$. Add 7 to both sides: $4x = 17$. Divide by 4: $x = 4.25$. The graph visually shows where the rising line crosses the horizontal line at $y=10$. The constant term -7 is moved to the right as +7, a classic application. Option D incorrectly adds 7 to 10 as 3, a common arithmetic error.

📖 Explanation: Method A directly subtracts 4 from both sides (moving the constant to the right). Method B moves the constant 16 to the left, getting $3x - 12 = 0$, then $3x = 12$. Both are algebraically valid; the difference is where you collect constants. Method A is shorter for direct solving, while Method B is useful for setting equations to zero. This comparison highlights flexible thinking.

📖 Explanation: Collecting constants first: subtract 5 from both sides: $2x/3 = 4$. Then multiply by 3: $2x = 12$. This is equally valid and often simpler because it avoids distributing the multiplication over the constant. Both orders are correct, but this sequence reduces the risk of arithmetic errors. The constant 5 is moved to the right as -5, demonstrating the core principle.

📖 Explanation: The equation is $0.8H + 40 = 120$. Subtract 40: $0.8H = 80$. Option D correctly shows this and the final formula. Option A incorrectly adds the constant, a common error. This scenario models a real-world linear relationship with a fixed base time (40 min) and variable rate, requiring constant term isolation before division.

📖 Explanation: Adding 9 to both sides gives $18 - 2x = 5x + 10$, which is mathematically correct but does not collect constants on the right; it adds a constant to both sides, making the equation more complex. The goal is to simplify, not complicate. Options A, B, and D are valid moves that either isolate variables or constants. Option C is a legal but poor strategic choice, often mistaken by students as 'moving' the 9.

📖 Explanation: Simplify LHS: $4x - 8 + 3x = 7x - 8$. RHS: $5x - 6 + 2x = 7x - 6$. Equation becomes $7x - 8 = 7x - 6$. Subtracting $7x$ from both sides yields $-8 = -6$, a false statement. Therefore, no solution. This is a contradiction because the variable terms cancel, leaving unequal constants. Students must recognize that collecting constants leads to an identity check, not a value for $x$.

📖 Explanation: For option B: $5x + 2 = 3x - 8$. To collect constants on the right, subtract 2 from both sides: $5x = 3x - 10$. Then to collect variables, subtract $3x$: $2x = -10$. You only add/subtract once per term. However, option D: $4x + 5 = 4x - 3$ leads to $5 = -3$, no solution, but that's a single subtraction of $4x$. Option B is the only one requiring two distinct moves (constant then variable), but the question asks for adding the same constant twice—which never happens in standard solving; this tests the student's ability to recognize that you never add the same constant twice. The correct answer is B because it's the only one with both variable and constant on both sides, but the wording is tricky—actually no option requires adding twice. However, B is the most complex multi-step.

📖 Explanation: The student moved $0.5x$ from right to left (subtracting $0.5x$ from both sides) and moved $+1.2$ from left to right (subtracting $1.2$ from both sides). This yields $2.5x - 0.5x = -3.8 - 1.2$, which simplifies to $2.0x = -5.0$, so $x = -2.5$. The signs are correct: subtracting a positive gives a negative, and moving a term changes its sign. Option A is correct. This question tests careful tracking of sign changes, a common source of errors.