Question

Zack used these steps to solve this equation:

Three-fourths y + 5 = negative three-fourths y minus 9. Step 1: StartFraction 6 Over 4 EndFraction y + 5 = negative 9. Step 2: StartFraction 6 Over 4 EndFraction y = negative 14. Step 3: (StartFraction 6 Over 4 EndFraction) StartFraction 6 Over 4 EndFraction y = negative 14 (StartFraction 6 Over 4 EndFraction). Step 4: y = negative 21.

When Zack checked his solution, it didn’t work. What was his mistake?
In step 1, he should have used the multiplication property of equality.
In step 2, he should have a constant of – 4 on the right side.
In step 3, he should have divided both sides by StartFraction 6 Over 4 Endfraction.
In step 4, the answer should be positive.

Answer 1

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 \(\frac{6}{4}\).**