Caleb solved this equation and recorded his work.

Question

7.4x + 4.1(2x − 4) = −2.3(x − 6) − 21.6
1. 7.4x + 8.2x − 16.4 = −2.3x + 13.8 − 21.6
2. 15.6x − 16.4 = −2.3x + 35.4
3. 17.9x − 16.4 = 35.4
4. 17.9x = 51.8
5. x ≈ 2.89
When Caleb verified his solution, it didn’t work. What mistake did he make?
In step 1, the distributive property was not used properly on the right side of the equation.
In step 2, the like terms were not combined properly on the right side of the equation.
In step 3, the addition property of equality was not used properly to isolate the variable term.
In step 5, the division property of equality was not used properly to solve for x.

Answer 1

To determine where Caleb made his mistake, let's review the steps he took and verify the calculations.

Starting with the original equation: \[ 7.4x + 4.1(2x - 4) = -2.3(x - 6) - 21.6 \]

Step 1: Distributing on the left side: \[ 4.1(2x - 4) = 4.1 \cdot 2x - 4.1 \cdot 4 = 8.2x - 16.4 \] So, the left side becomes: \[ 7.4x + 8.2x - 16.4 \] On the right side: \[ -2.3(x - 6) = -2.3x + 13.8 \] So, the right side becomes: \[ -2.3x + 13.8 - 21.6 \] This simplifies to: \[ -2.3x - 7.8 \]

Combining these gives us: \[ 7.4x + 8.2x - 16.4 = -2.3x - 7.8 \]

Step 2: Combining like terms on the left gives: \[ 15.6x - 16.4 = -2.3x - 7.8 \] This is where we notice a correction from Caleb's work.

Returning to his original Step 2, he incorrectly stated: \[ 15.6x - 16.4 = -2.3x + 35.4 \] The correct calculation should have resulted in: \[ 15.6x - 16.4 = -2.3x - 7.8 \]

Step 3: At this point, the error leads to incorrect conclusions in subsequent steps, but we can already identify the issue.

The correct line should have been: \[ 15.6x - 16.4 = -2.3x - 7.8 \] He did not combine like terms properly on the right side of the equation.

Therefore, the answer is: In step 2, the like terms were not combined properly on the right side of the equation.