📝 How to verify a solution to an equation (14 MCQs)

📖 Explanation: Efficient verification requires comparing the simplified numerical values of both sides after substitution. Option C is the only one that checks the entire equation. Substituting into only parts (A, B, D) is incomplete and cannot disprove the equality because errors in other terms might cancel or be missed, leading to a false conclusion.

📖 Explanation: Since the simplified equation $4(2y-1)=y+15$ is mathematically equivalent to the original, substituting into either the original (B) or the simplified form (C) is valid. Option A is also correct but redundant. Option D correctly identifies that both B and C are acceptable verification methods, demonstrating understanding that simplification preserves equality.

📖 Explanation: While equal values confirm the equality at $x=4$, verification must also ensure that the solution does not make any denominator zero or violate the domain of the original equation. Maria’s check is necessary but not sufficient for equations with variables in denominators. Option D highlights this critical, often-overlooked step in verification, which is a higher-order error analysis.

📖 Explanation: The intersection point $(3,4)$ means that when $x=3$, both expressions yield $y=4$. Therefore, $x=3$ satisfies the equation $3x-5 = -2x+10$. Option A correctly interprets the graph. Options B and C confuse the x and y coordinates of the intersection point, a common misconception. Option D is false because intersection implies a solution exists.

📖 Explanation: Bob’s method is the most rigorous because he explicitly computes both sides numerically and compares them. Carol’s statement is vague without computation. Alice relies on her solving process, which may have errors. David’s graph is a visual approximation, not an exact algebraic verification unless exact coordinates are given. Bob provides concrete, verifiable numerical evidence, making his method the strongest.

📖 Explanation: While the substitution works, the student must verify that the denominator $x-1$ is not zero at $x=2.5$. Here, $2.5-1=1.5 \neq 0$, so it's safe. However, the omission of this check is a critical error in verification, especially for rational equations. Option C correctly identifies this missing step, emphasizing that domain restrictions are part of solution verification, not just numerical equality.

📖 Explanation: Equation (II) is exactly 2 times Equation (I). Since multiplying an equation by a non-zero constant produces an equivalent equation, any solution of (I) is also a solution of (II). Option A demonstrates this equivalence property. Options B, C, and D reflect misconceptions about equivalent equations, failing to recognize the scalar multiple relationship, which is a key algebraic concept.

📖 Explanation: The student correctly substituted $x=-2$ into the absolute value expression and obtained 1, which does not equal 3. Therefore, $x=-2$ is indeed not a solution. Option A is correct. Options B and C are distractors: while $|x+1|=3$ has solutions $x=2$ and $x=-4$, that doesn't make $x=-2$ a solution. Option D is false because the evaluation $|-1|=1$ is correct. This tests whether students understand that verification is a definitive test.

📖 Explanation: Options A, B, and C are all valid verification methods because they maintain the equality or compare equivalent forms. Option D is invalid because it ignores the constant term +4 on the left side, changing the equation. Substituting into only parts of an expression is a common error. This question requires students to analyze the validity of different verification procedures and identify an incomplete or incorrect method, testing error analysis skills.

📖 Explanation: Multiplying an equation by 0 yields the identity $0=0$, which is true for all $x$, including $x=3$. However, this operation is not reversible (it loses information), so the new equation is not equivalent to the original. While $x=3$ satisfies $0=0$, it cannot be verified as a solution to the original equation using this transformed equation. Option D correctly distinguishes between 'satisfying the new equation' and 'being a solution to the original', a subtle and advanced concept.

📖 Explanation: Verification serves two purposes: (1) it checks for arithmetic errors made during solving (Option A), and (2) it ensures the final value satisfies the original equation, especially when operations like multiplying by a variable or squaring might introduce extraneous solutions (Option C). Option D correctly combines both, showing a comprehensive understanding of why verification is essential, beyond just a routine check.

📖 Explanation: The equation $2x+1 = -x+4$ represents the x-coordinate of the intersection of the two lines. Substituting $x=1$ gives $3=3$, verifying the equality (Option A). Graphically, this means the point (1,3) lies on both lines, confirming the intersection (Option B). Option C correctly combines both interpretations. This question requires students to connect algebraic verification with graphical meaning, testing mixed concepts and graph interpretation.

📖 Explanation: Verification is not a proof of the solution's uniqueness or derivation; it's a confirmation that the proposed value satisfies the equation. The first student is correct that the check works. The second student raises a valid philosophical point: verification doesn't prove the solving steps were correct, but it does provide evidence. Option C captures this nuance: verification is a necessary (but not sufficient) check for correctness, not a circular argument. This tests deep understanding of the role of verification in mathematics.

📖 Explanation: Squaring both sides of an equation can introduce extraneous solutions that satisfy the squared equation but not the original. Here, $x=-1$ satisfies $(\sqrt{2x+3})^2 = x^2$ but not the original because the principal square root is non-negative, so $\sqrt{1}=1 \neq -1$. Option B is the algebraic reason, and Option C is the domain/logical reason. Option D correctly combines both, requiring students to understand the underlying cause of the failed verification, not just the procedural check.