Petro was given this system of equations.

Question

Negative 14 x minus 2 y = 24. 14 x + 8 y = negative 12.

Petro’s work is shown in the table. Where, if anywhere, did Petro first make a mistake?

Steps
Petro’s Work

Step 1

Negative 14 x minus 2 y = 24. 14 x + 8 y = negative 12. 6 y = 12.

Step 2

6 y = 12. y = 2.

Step 3

Negative 14 (2) minus 2 y = 24. Negative 28 minus 2 y = 24. Negative 2 y = 52. y = negative 26.

step 1
step 2
step 3
no mistake

Answer 1

Let's examine Petro's work step by step to identify where any mistakes may have occurred.

Step 1:

The equations are:

1. \(-14x - 2y = 24\)
2. \(14x + 8y = -12\)

Petro states "6y = 12." However, this equation does not correctly represent either original equation. Therefore, the first mistake is made here.

Step 2:

Petro correctly solves for \(y\) from the equation \(6y = 12\): \[ y = \frac{12}{6} = 2. \]

This step is correct, but it started from a wrong equation.

Step 3:

In this step, Petro substitutes \(y = 2\) into the first equation \(-14(2) - 2y = 24\) correctly, yielding: \[-28 - 2y = 24.\] He correctly simplifies to find: \[-2y = 52,\] and then finds: \[y = -26.\]

However, since Petro made a mistake in Step 1, all subsequent calculations are based on incorrect assumptions.

Conclusion:

Petro's first mistake was in Step 1 where he incorrectly simplified or manipulated the equations, ending up with \(6y = 12\), which does not follow from either original equation. Thus, the mistake first occurs in Step 1.