Question

use the following information to answer questions 15-16.
lin is solving the system of equations.
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$$\begin{cases} 6x - 5y = 34 \\\\ 3x + 2y = 8 \\end{cases}$$

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she starts by rearranging the second equation to isolate the \\( y \\)-variable, \\( y = 4 - 1.5x \\).
she then substituted the expression \\( 4 - 1.5x \\) for \\( y \\) in the first equation as shown:
\\( \text{step 1: } 6x - 5(4 - 1.5x) = 34 \\) \\( \text{step 5: } y = 4 - 1.5x \\)
\\( \text{step 2: } 6x - 20 - 7.5x = 34 \\) \\( \text{step 6: } y = 4 - 1.5 \cdot (-16) \\)
\\( \text{step 3: } -1.5x = 54 \\) \\( \text{step 7: } y = 30 \\)
\\( \text{step 4: } x = -16 \\)

1. in which step did lin write the equation that contained the error that caused the system to be solved incorrectly?

\\( \text{a) step 1} \\)
\\( \text{b) step 2} \\)
\\( \text{c) step 3} \\)
\\( \text{d) step 6} \\)

1. solve lins system of equations correctly.

\\\

$$\begin{cases} 6x - 5y = 34 \\\\ 3x + 2y = 8 \\end{cases}$$

\\
\\( x = \square \\)
\\( y = \square \\)

Explanation:

Question 15

To determine the step with the error, we analyze each step:

• Step 1: Substitute \( y = 4 - 1.5x \) into \( 6x - 5y = 34 \), so \( 6x - 5(4 - 1.5x) = 34 \). This is correct.
• Step 2: Distribute \(-5\) into \( (4 - 1.5x) \). The correct distribution is \( 6x - 20 + 7.5x = 34 \) (since \(-5 \times -1.5x = +7.5x\)), but Lin wrote \( 6x - 20 - 7.5x = 34 \). This is the error.
• Step 3, 4, 6, 7: Depend on Step 2, so the error starts in Step 2.

Step 1: Solve the second equation for \( y \)

The second equation is \( 3x + 2y = 8 \). Subtract \( 3x \) from both sides: \( 2y = 8 - 3x \). Divide by 2: \( y = 4 - 1.5x \).

Step 2: Substitute \( y \) into the first equation

First equation: \( 6x - 5y = 34 \). Substitute \( y = 4 - 1.5x \):
\( 6x - 5(4 - 1.5x) = 34 \)
Distribute \(-5\): \( 6x - 20 + 7.5x = 34 \)
Combine like terms: \( 13.5x - 20 = 34 \)
Add 20 to both sides: \( 13.5x = 54 \)
Divide by \( 13.5 \): \( x = \frac{54}{13.5} = 4 \)

Step 3: Find \( y \) using \( x = 4 \)

Substitute \( x = 4 \) into \( y = 4 - 1.5x \):
\( y = 4 - 1.5(4) = 4 - 6 = -2 \)

Answer:

b) Step 2

Question 16