Asked by Makayla
While grading her students' most recent quiz on equation solving, Mrs. Jones calculated that approximately forty percent of her students answered question number 14 with multiple-choice option B, while the other sixty percent answered A or C.
Question #14 from Mrs. Jones's students' most recent quiz:
Solve the single variable equation for n.
A. n = 3
B. No solution
C. Infinitely many solutions
Use complete sentences to compare the similarities and differences of each of the multiple-choice answer options A-C. In your answer, rationalize why a student would choose each of the options as the correct answer.
All Answers 1
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GPT-5 mini
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All three answer choices describe possible outcomes when you solve a single-variable equation, but they differ in the size and nature of the solution set. Choice A (n = 3) asserts a unique, single solution. Choice B (No solution) asserts the solution set is empty — no value of n satisfies the equation. Choice C (Infinitely many solutions) asserts every (or infinitely many) values of n satisfy the equation so the solution set is not finite. The three choices are mutually exclusive: an equation cannot simultaneously have exactly one solution, no solutions, and infinitely many solutions.
Why a student would choose A (n = 3). A student who correctly isolates the variable and obtains a single value will choose A. For example, solving 2n − 6 = 0 gives 2n = 6 and n = 3, so A is correct in that case. A student might also arrive at n = 3 by algebraic simplification of a more complicated equation that legitimately reduces to that single value.
Why a student would choose B (No solution). A student will choose B if simplification leads to a contradiction (an impossible statement) such as 0 = 5. For example, starting from 2(n − 3) = 2n − 5, expanding gives 2n − 6 = 2n − 5 and subtracting 2n from both sides gives −6 = −5, a contradiction, so no n satisfies the equation. A student might also pick B if they correctly see that every attempt to satisfy the equation violates a domain restriction (for instance, every candidate would make a denominator zero), or if they made an algebraic error that produced an impossible equality.
Why a student would choose C (Infinitely many solutions). A student will choose C if simplification leads to an identity (a true statement that does not involve the variable), for example 0 = 0. For instance, 2(n − 3) = 2n − 6 expands to 2n − 6 = 2n − 6, which reduces to 0 = 0; any real n makes the original equality true, so there are infinitely many solutions. A student might also choose C if they cancel a variable and conclude the variable disappears leaving a true identity, or if they misunderstand domain restrictions and think “variable cancels so everything works.”
Common student errors that explain mixed responses. When students manipulate expressions they can make mistakes that lead to different apparent outcomes. Canceling a common factor without checking whether that factor could be zero can cause a student to lose a valid solution (leading them to an incorrect B or to miss A). Dividing by an expression that might be zero can produce extraneous or lost solutions. Squaring both sides, expanding incorrectly, sign errors, or arithmetic slips can turn a correct single-solution problem into a false contradiction (B) or into an apparent identity (C). Because of these possible correct results and common mistakes, different students can reasonably arrive at A, B, or C depending on their algebra and checks for domain issues.
Why a student would choose A (n = 3). A student who correctly isolates the variable and obtains a single value will choose A. For example, solving 2n − 6 = 0 gives 2n = 6 and n = 3, so A is correct in that case. A student might also arrive at n = 3 by algebraic simplification of a more complicated equation that legitimately reduces to that single value.
Why a student would choose B (No solution). A student will choose B if simplification leads to a contradiction (an impossible statement) such as 0 = 5. For example, starting from 2(n − 3) = 2n − 5, expanding gives 2n − 6 = 2n − 5 and subtracting 2n from both sides gives −6 = −5, a contradiction, so no n satisfies the equation. A student might also pick B if they correctly see that every attempt to satisfy the equation violates a domain restriction (for instance, every candidate would make a denominator zero), or if they made an algebraic error that produced an impossible equality.
Why a student would choose C (Infinitely many solutions). A student will choose C if simplification leads to an identity (a true statement that does not involve the variable), for example 0 = 0. For instance, 2(n − 3) = 2n − 6 expands to 2n − 6 = 2n − 6, which reduces to 0 = 0; any real n makes the original equality true, so there are infinitely many solutions. A student might also choose C if they cancel a variable and conclude the variable disappears leaving a true identity, or if they misunderstand domain restrictions and think “variable cancels so everything works.”
Common student errors that explain mixed responses. When students manipulate expressions they can make mistakes that lead to different apparent outcomes. Canceling a common factor without checking whether that factor could be zero can cause a student to lose a valid solution (leading them to an incorrect B or to miss A). Dividing by an expression that might be zero can produce extraneous or lost solutions. Squaring both sides, expanding incorrectly, sign errors, or arithmetic slips can turn a correct single-solution problem into a false contradiction (B) or into an apparent identity (C). Because of these possible correct results and common mistakes, different students can reasonably arrive at A, B, or C depending on their algebra and checks for domain issues.
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