Let's analyze Zack's steps to identify the mistake he made while solving the equation:
The original equation is: \[ \frac{3}{4}y + 5 = -\frac{3}{4}y - 9 \]
Step 1:
Zack states: \[ \frac{6}{4}y + 5 = -9 \] To clarify: The left side should just be simplified by combining like terms from the original equation correctly. From the original equation, if we add \(\frac{3}{4}y\) to both sides, we'll actually have: \[ \frac{3}{4}y + \frac{3}{4}y + 5 = -9 \] which simplifies to: \[ \frac{6}{4}y + 5 = -9 \] This step itself is acceptable.
Step 2:
Zack continues with: \[ \frac{6}{4}y = -14 \] Here, he correctly subtracts 5 from both sides: \[ \frac{6}{4}y = -9 - 5 \Rightarrow \frac{6}{4}y = -14 \] This step is correct.
Step 3:
Zack writes: \[ \left(\frac{6}{4}\right)\left(\frac{6}{4}\right)y = -14\left(\frac{6}{4}\right) \] The way this step is executed is not necessary. Instead, he should divide by \(\frac{6}{4}\) (or multiply by its reciprocal) to isolate y.
Step 4:
In this step, he calculates: \[ y = -21 \] Since we haven't checked his multiplication in the previous step, let's calculate what \(y\) should actually be.
To solve for \(y\) correctly, to get from Step 2 to Step 4, we should divide \( -14 \) by \( \frac{6}{4} \): \[ y = -14 \cdot \frac{4}{6} \] Calculating this gives: \[ y = -14 \cdot \frac{2}{3} = -\frac{28}{3} \]
Thus, the real correct answer for \(y\) isn’t -21; it should have been \(-\frac{28}{3}\).
Conclusion:
The mistake is in Step 3 where he incorrectly multiplied rather than dividing properly by \(\frac{6}{4}\) to isolate \(y\). Therefore, the correct assessment is:
In Step 3, he should have divided both sides by \(\frac{6}{4}\).