Question

sample response: the equation \\(ax = bx + 1\\) is the same as \\(x = 1/(a - b)\\) when solved for \\(x\\). this means that \\(x\\) is equal to the reciprocal of the difference of \\(a\\) and \\(b\\).

what did you include in your response? check all that apply.

\\(x = \frac{1}{a - b}\\)
\\(x\\) is the reciprocal of the difference of \\(a\\) and \\(b\\).
\\(x\\) is 1 over the difference of \\(a\\) and \\(b\\).
\\(x\\) is the quotient of 1 and the difference of \\(a\\) and \\(b\\).

Explanation:

Analyze the sample response

Using the Textual Comparison and Error Analysis knowledge points, we examine the provided text:
"The equation \(ax = bx + 1\) is the same as \(x = 1/(a - b)\) when solved for x. This means that x is equal to the reciprocal of the difference of a and b."

We must check which of the given options are explicitly included or directly stated in this sample response.

Evaluate the first option

Using the Literal Equations and Algebraic Rearrangement knowledge points

$$ x = \frac{1}{a - b} $$

The sample response states: "the same as \(x = 1/(a - b)\)". This is mathematically identical to the written fraction. Thus, this is included.

Evaluate the second option

Using the Mathematical Translation knowledge point, we compare the text:
"x is the reciprocal of the difference of a and b."
The sample response states: "This means that x is equal to the reciprocal of the difference of a and b." This matches word-for-word. Thus, this is included.

Evaluate the third option

Using the Error Analysis and Textual Comparison knowledge points, we check:
"x is 1 over the difference of a and b."
Although mathematically equivalent to the equation, the phrase "1 over" does not appear anywhere in the sample response text. The prompt asks "What did you include in your response?", which refers to the literal elements or phrases written in the sample response. Therefore, this option is not included.

Evaluate the fourth option

Using the Mathematical Translation and Textual Comparison knowledge points, we check:
"x is the quotient of 1 and the difference of a and b."
The phrase "quotient of 1 and..." is not written in the sample response. Therefore, this option is not included.

Answer:

• (A) \(x = \frac{1}{a - b}\) (Correct answer)
• (B) \(x\) is the reciprocal of the difference of \(a\) and \(b\). (Correct answer)
• (C) \(x\) is 1 over the difference of \(a\) and \(b\).
• (D) \(x\) is the quotient of 1 and the difference of \(a\) and \(b\).