Question

a student is solving the equation \\(\frac{3}{x - 1} = \frac{4}{2x + 1}\\). the table shows the student’s steps.\
\

step	equation	\
---	---	\
given	\\(\frac{3}{x - 1} = \frac{4}{2x + 1}\\)	\
1	\\(\frac{3}{x - 1} \cdot (x - 1)(2x + 1) = \frac{4}{2x + 1} \cdot (x - 1)(2x + 1)\\)	\
2	\\(3(x - 1) = 4(2x + 1)\\)	\
3	\\(3x - 3 = 8x + 4\\)	\
4	\\(-5x - 3 = 4\\)	\
5	\\(-5x = 7\\)	\
6	\\(x = -\frac{7}{5}\\)	\

\
which statement is true?\
\

• the student made an error simplifying between step 1 and step 2\
• the student made an error distributing between step 2 and step 3\
• the student made an error subtracting between step 3 and step 4\
• the student made an error dividing between step 5 and step 6

Explanation:

Step1: Check Step1 to Step2 simplification

When multiplying $\frac{3}{x-1} \cdot (x-1)(2x+1)$, the $(x-1)$ terms cancel, leaving $3(2x+1)$. When multiplying $\frac{4}{2x+1} \cdot (x-1)(2x+1)$, the $(2x+1)$ terms cancel, leaving $4(x-1)$. The student incorrectly wrote $3(x-1)=4(2x+1)$.

Step2: Verify other steps for accuracy

• Step2 to Step3: Distributing $3(x-1)$ gives $3x-3$, distributing $4(2x+1)$ gives $8x+4$; this step is correct if Step2 was right.
• Step3 to Step4: Subtract $8x$ from both sides: $3x-8x-3=4 \implies -5x-3=4$; this is correct.
• Step5 to Step6: Divide both sides by $-5$: $x=-\frac{7}{5}$; this is correct.

Answer:

The student made an error simplifying between Step 1 and Step 2