Henry showed all the steps he took to solve the equation.

Question

4 (negative one-half x + 7) + 5 x = 15

Step
Simplified Expression
1
-Negative 2 x + 28 + 5 x = 15
2
3 x + 28 = 15
3
3 x = 15
4
x = 5

Sammy tried to verify the solution by substituting x = 5 back into the original equation. It did not result in a true statement. Which explains why Henry’s solution is incorrect?
Step 1 is wrong because the distributive property of equality was used incorrectly.
Step 2 is wrong because the addition property of equality was used incorrectly.
Step 3 is wrong because the subtraction property of equality was used incorrectly.
Step 4 is wrong because the division property was used incorrectly.

Answer 1

Let's analyze the steps from Henry's solution to identify where he went wrong.

The original equation is: \[ 4 \left(-\frac{1}{2} x + 7\right) + 5x = 15 \]

Step 1:

Henry writes: \[ -2x + 28 + 5x = 15 \]

The left side should be calculated using the distributive property: \[ 4 \left(-\frac{1}{2} x + 7\right) = 4 \cdot -\frac{1}{2}x + 4 \cdot 7 = -2x + 28 \] So, this step is correct, as Henry applied the distributive property correctly.

Step 2:

Henry then writes: \[ 3x + 28 = 15 \]

He combines \(-2x\) and \(5x\) correctly: \[ -2x + 5x = 3x \]

This step is also correct.

Step 3:

Henry writes: \[ 3x = 15 - 28 \] From here, Henry simplifies incorrectly. The correct operation would be: \[ 3x + 28 = 15 \implies 3x = 15 - 28 \implies 3x = -13 \] So this step is incorrect as he did not subtract 28 correctly from 15.

Step 4:

Henry then writes: \[ x = 5 \]

Since he derived \(3x = 15\), he would get \(x = 5\) based on the incorrect earlier step.

In conclusion, the error arises in Step 3 where the subtraction from 15 was handled incorrectly, leading to Henry's incorrect solution.

The correct explanation for why Henry’s solution is incorrect is:

Step 3 is wrong because the subtraction property of equality was used incorrectly.