📝 General Strategy for Solving Linear Equations (14 MCQs)

📖 Explanation: This equation simplifies to $3x-12+2x = 5x-12$, which becomes $5x-12=5x-12$. This is an identity, meaning it is true for all real numbers, not just $x=0$. The student made a fundamental error by stopping after finding one solution, failing to recognize the identity. This common mistake stems from treating an identity like a conditional equation.

📖 Explanation: Let $x$ represent the number of miles. The equation is $50+0.20x = 30+0.30x$. Solving: subtract $0.20x$ from both sides to get $50 = 30+0.10x$, then $20=0.10x$, so $x=200$. This models a real-world break-even point. Distractors often come from arithmetic errors or misinterpreting the fixed and variable costs.

📖 Explanation: Equation B simplifies to $3x-3=3x-2$. Subtracting $3x$ from both sides gives $-3=-2$, a false statement. This indicates no solution. Equations A and C are identities (true for all $x$), while equation D is conditional and has a single solution. This tests the student's ability to distinguish between identities, contradictions, and conditional equations.

📖 Explanation: The intersection point of two linear functions represents the value of $x$ for which $f(x) = g(x)$. The x-coordinate of the intersection is the solution to the equation. The y-coordinate is the value of the functions at that point. Distractor C confuses the output with the input. Distractor A ignores the fact that intersection implies a solution.

📖 Explanation: The student's sequence is mathematically sound. They can subtract 4 first to get $\frac{2x}{3} = \frac{x}{2} - 5$, then multiply by 6 to get $4x = 3x - 30$. This leads to $x=-30$. There is no error. The distractor B suggests a common misconception that the order of operations is fixed, but in equation solving, you can perform operations in any order as long as you maintain equality.

📖 Explanation: Multiplying both sides by 4 clears the decimals: $(4)(0.25)(4x-8) = (4)(0.5x+2)$ simplifies to $1(4x-8)=2x+8$. This is the most efficient because it immediately eliminates decimals. Distributing first (C) is also valid but leads to more complex arithmetic with decimals. Dividing by 0.25 (A) is a valid but less intuitive first step. This question tests strategic thinking in equation solving.

📖 Explanation: The least common denominator is the least common multiple (LCM) of the denominators. The LCM of 2, 3, and 4 is 12. Using the LCD is efficient because it results in the smallest possible integers. Distractor B (24) is a common multiple but not the least, leading to larger numbers and more potential arithmetic errors. Using a non-least common denominator is not wrong but is inefficient.

📖 Explanation: Method B: divide by 2 to get $(x-3)+2x = 3(x-1)$, which simplifies to $3x-3 = 3x-3$, an identity. Method A: $2x-6+4x = 6x-6$, which also simplifies to $6x-6=6x-6$. Both lead to the identity $x=x$, meaning infinite solutions. Method B is more efficient because it reduces the numbers before distributing. This tests the ability to compare solution strategies and recognize efficiency.

📖 Explanation: The equation simplifies to $2x+5 = 2x+6$. Subtracting $2x$ from both sides yields $5=6$, which is a contradiction. Therefore, the equation has no solution. The student's claim that $x=1$ is a solution can be checked: $2(1)+5=7$ and $2(1+3)=8$, which are not equal. This question tests the ability to verify a proposed solution and identify a contradiction.

📖 Explanation: Solving $2x+3=7$ gives $2x=4$, so $x=2$. The solution to a linear equation in one variable is a single value, which is represented as a single point on the number line. Distractor B is a common sign error. Distractors C and D represent inequalities or intervals, which are not solutions to a single equation. This tests the fundamental understanding of representing solutions graphically.

📖 Explanation: The width is $w$. The length is $2w+3$. The perimeter of a rectangle is $2(\text{length} + \text{width}) = 2(w + (2w+3)) = 2w + 2(2w+3)$. Option A correctly applies the perimeter formula. Option B is the perimeter of a triangle. Option C is the sum of the length and width without the factor of 2. Option D incorrectly distributes the 2. This models a real-world geometry problem.

📖 Explanation: LCD is 12. Multiply by 12: $3(x+2) - 2(2x-1) = 4$. Simplify: $3x+6 - 4x + 2 = 4$, which gives $-x + 8 = 4$, so $-x = -4$, $x=4$. This problem involves multiple fractions, distribution of negative signs, and careful arithmetic. The distractors come from common errors like incorrect LCD, sign errors when distributing the negative, or errors in combining like terms. This is a multi-step reasoning problem that tests precision.

📖 Explanation: For an equation to have infinitely many solutions, both sides must be identical expressions. Expanding the left side gives $6x-15$. Therefore, for the equation to be an identity, $a$ must equal 6 and $b$ must equal $-15$. Distractor B has the wrong sign for $b$. Distractor C incorrectly copies the coefficients from the original expression. This tests the deep understanding of the conditions for an identity in a parametric context.

📖 Explanation: Multiplying by the reciprocal $\frac{3}{2}$ and dividing by the fraction $\frac{2}{3}$ are mathematically equivalent operations. Both yield $x=8 \cdot \frac{3}{2} = 12$. This question tests the understanding of inverse operations and their equivalence. Distractor B is a common misconception about reciprocals. Distractor C incorrectly states that division by a fraction is not allowed.