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Lin is solving this system of equations:
[tex]\[
\begin{array}{l}
6x - 5y = 34 \\
3x + 2y = 8
\end{array}
\][/tex]

She starts by rearranging the second equation to isolate the [tex]\(y\)[/tex] variable: [tex]\(y = 4 - 1.5x\)[/tex]. She then substitutes the expression [tex]\(4 - 1.5x\)[/tex] for [tex]\(y\)[/tex] in the first equation, as shown:

[tex]\[
\begin{array}{l}
6x - 5(4 - 1.5x) = 34 \\
6x - 20 + 7.5x = 34 \\
13.5x - 20 = 34 \\
13.5x = 54 \\
x = 4 \\
y = 4 - 1.5x \\
y = 4 - 1.5(4) \\
y = 4 - 6 \\
y = -2
\end{array}
\][/tex]

Lin made a mistake in her work. Find and explain what Lin's mistake was.



Answer :

GinnyAnswer
Certainly! Let's analyze Lin's steps and identify where her mistake occurred.

### Step-by-Step Analysis

1. Initial System of Equations:
[tex]\[ \begin{array}{l} 6x - 5y = 34 \quad \text{(Equation 1)} \\ 3x + 2y = 8 \quad \text{(Equation 2)} \end{array} \][/tex]

2. Isolate [tex]\(y\)[/tex] using Equation 2:
[tex]\[ \begin{aligned} 3x + 2y &= 8 \\ 2y &= 8 - 3x \\ y &= 4 - 1.5x \end{aligned} \][/tex]
Lin correctly rearranges Equation 2 to isolate [tex]\(y\)[/tex].

3. Substitute [tex]\(y = 4 - 1.5x\)[/tex] into Equation 1:
[tex]\[ 6x - 5(4 - 1.5x) = 34 \][/tex]
Now distribute the [tex]\(-5\)[/tex]:
[tex]\[ \begin{aligned} 6x - 5 \cdot 4 + 5 \cdot 1.5x &= 34 \\ 6x - 20 + 7.5x &= 34 \end{aligned} \][/tex]
Combining like terms:
[tex]\[ 13.5x - 20 = 34 \][/tex]

4. Solve for [tex]\(x\)[/tex]:
[tex]\[ \begin{aligned} 13.5x - 20 &= 34 \\ 13.5x &= 54 \\ x &= \frac{54}{13.5} \\ x &= 4 \end{aligned} \][/tex]

5. Find [tex]\(y\)[/tex] using [tex]\(x = 4\)[/tex]:
[tex]\[ \begin{aligned} y &= 4 - 1.5 \cdot 4 \\ y &= 4 - 6 \\ y &= -2 \end{aligned} \][/tex]

### Lin's Mistake
Lin made a mistake while simplifying the expression [tex]\(6x - 5(4 - 1.5x)\)[/tex].

Here is the correct simplification:
[tex]\[ 6x - 5(4 - 1.5x) = 6x - 20 + 7.5x = 13.5x - 20 \][/tex]

### Lin's Incorrect Simplification
Lin incorrectly simplified it as:
[tex]\[ 6x - 5(4 - 1.5x) = 6x - 20 - 7.5x \][/tex]
This led her to:
[tex]\[ -1.5x - 20 = 34 \quad (\text{Incorrect}) \][/tex]

### Correct Values
The correct values after properly simplifying and solving are:
[tex]\[ x = 4 \quad \text{and} \quad y = -2 \][/tex]

### Conclusion
Lin’s mistake was in the simplification step while substituting [tex]\( y = 4 - 1.5x \)[/tex] into the first equation. She incorrectly included a negative sign in front of [tex]\( 7.5x \)[/tex], leading to erroneous values for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. The correct solutions are [tex]\( x = 4 \)[/tex] and [tex]\( y = -2 \)[/tex].

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