📝 Solve equations with constants on both sides (14 MCQs)

📖 Explanation: The student's steps are mathematically correct. Subtracting $3x$ isolates the variable term on the right, adding 3 isolates the constant, and dividing by 2 gives $x=5$. A common misconception is that the order of operations is wrong, but here the inverse operations are applied correctly in sequence.

📖 Explanation: The original equation simplifies to $2x = 16$ after adding 9 and subtracting $2x$. Option A is exactly that simplified form. Option B is equivalent but not simplified; option C has incorrect sign; option D ignores the $4x$ term. Recognizing equivalent forms tests understanding of equation transformation.

📖 Explanation: The perimeter formula is $2(L+W)=48$. Substituting gives $2(3x+2)=48 \Rightarrow 6x+4=48 \Rightarrow 6x=44 \Rightarrow x=7.33$

📖 Explanation: Actually $7 - 3(-1) = 7+3=10$ and $4(-1)+14 = -4+14=10$, so $x=-1$ is correct. The question tests if student can verify by substitution. Many students might make sign errors when moving $-3x$ or adding constants, but here the solution is valid.

📖 Explanation: Setting $5x - 2 = 3x + 8$ gives $2x = 10 \Rightarrow x=5$. Substituting into either equation gives $y=5(5)-2=23$. So point is $(5,23)$. This tests the connection between equation solving and intersection of linear graphs, a key conceptual bridge.

📖 Explanation: Subtract $0.10t$ from both sides: $20 = 12 + 0.05t$. Subtract 12: $8 = 0.05t$. Divide by 0.05: $t = 160$. This models a real-world break-even scenario. A common mistake is to subtract 20 first, leading to $0.10t = -8 + 0.15t$ then $-0.05t = -8 \Rightarrow t=160$ still correct, but sign errors are frequent.

📖 Explanation: Method A: $9 = 4x - 7 \Rightarrow 16=4x \Rightarrow x=4$. Method B: $2x+16 = 6x \Rightarrow 16=4x \Rightarrow x=4$. Both are valid; order of inverse operations doesn't change the solution. This tests understanding that equation solving is flexible as long as operations are legal.

📖 Explanation: Cross-multiply: $2(2x+5) = 3(x-1) \Rightarrow 4x+10 = 3x-3 \Rightarrow x = -13$. This involves fractions and constants on both sides, requiring distribution and careful integer arithmetic. A common error is to forget to distribute the 3 or 2 correctly.

📖 Explanation: The student moved all variable terms to left (subtract $5x$) and constants to right (add 3). That yields $3x = 12$ which is correct. Then dividing by 3 gives 4. This is a perfectly valid method. The distractor 'sign error' is common when students incorrectly add/subtract, but here it's done correctly.

📖 Explanation: Distribute correctly: $4(x-2) = 4x - 8$. Right side: $3(x+5) = 3x+15$, then subtract $2x$ gives $3x+15-2x = x+15$. So equation becomes $4x - 8 = x+15$. Option A shows the intermediate distribution step before combining like terms, which is correct. This tests the order of operations in multi-step equations.

📖 Explanation: Set $30 + 0.20m = 50$. Subtract 30: $0.20m = 20$. Divide by 0.20: $m=100$. This is a simpler case where constant on one side is zero after moving. Students might incorrectly set $30 = 50 - 0.20m$ and get negative sign errors, but the solution is straightforward.

📖 Explanation: Reciprocal of $-9$ is $-1/9$. Solving $\frac{3}{x} + 4 = \frac{2}{x} - 5$: multiply by $x$: $3+4x = 2 - 5x \Rightarrow 9x = -1 \Rightarrow x = -1/9$. This combines reciprocal concept with equation solving, testing deeper understanding of variable in denominator and inverse relationships.

📖 Explanation: Both methods use the addition property of equality (adding or subtracting the same quantity from both sides). The order doesn't matter because these properties allow us to perform any sequence of legal operations. This distinguishes between properties of numbers (commutative/associative) and properties of equations (equality properties).

📖 Explanation: After simplification, we have $6x - 2 = -2x - 2$. To isolate $x$, add $2x$ to both sides to get $8x - 2 = -2$, then add 2 to get $8x = 0 \Rightarrow x=0$. Option A is correct order. Option D would give $8x - 4 = -2$ which is unnecessary. This tests the strategic sequence of inverse operations.