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

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

a. Check to see if Lin's solution of [tex](4, -2)[/tex] makes both equations in the system true.

b. If your answer to the previous question is "no," find and explain her mistake. If your answer is "yes," graph the equations to verify the solution of the system.

Sagot :

GinnyAnswer
Let's tackle the problem step by step.

### Part A: Checking the Solution

1. First Equation Verification:

We are given the equations:
[tex]\[ 6x - 5y = 34 \][/tex]
[tex]\[ y = 4 - 1.5x \][/tex]

Lin claims that the solution is \((x, y) = (-36, 58)\).

Let's plug in \(x = -36\) and \(y = 58\) into the first equation:
[tex]\[ 6(-36) - 5(58) = 34 \][/tex]

Calculate \(6(-36)\):
[tex]\[ 6 \times -36 = -216 \][/tex]

Calculate \(-5(58)\):
[tex]\[ -5 \times 58 = -290 \][/tex]

Now sum these results:
[tex]\[ -216 - 290 = -506 \][/tex]

The left side of the equation is \(-506\), which does not equal \(34\). So the given solution fails to satisfy the first equation.

2. Second Equation Verification:

Now let's check the second equation:
[tex]\[ y = 4 - 1.5x \][/tex]

Substitute \(x = -36\):
[tex]\[ y = 4 - 1.5(-36) \][/tex]

Calculate \(1.5 \times -36\):
[tex]\[ 1.5 \times -36 = -54 \][/tex]

Now substitute back:
[tex]\[ y = 4 - (-54) \][/tex]
[tex]\[ y = 4 + 54 \][/tex]
[tex]\[ y = 58 \][/tex]

The left side of the equation is \(58\), which matches the value of \(y\). So, this solution satisfies the second equation.

### Part B: Analyzing the Mistake

Since the solution did not make both equations true, there is a mistake somewhere. Let's find Lin's mistake in her calculation:

Original system of equations:
[tex]\[ 6x - 5y = 34 \][/tex]
[tex]\[ y = 4 - 1.5x \][/tex]

Following Lin's steps:

- Substitute \(y = 4 - 1.5x\) into \(6x - 5y = 34\):
[tex]\[ 6x - 5(4 - 1.5x) = 34 \][/tex]
[tex]\[ 6x - 20 + 7.5x = 34 \][/tex]
[tex]\[ 6x + 7.5x = 34 + 20 \][/tex]
[tex]\[ 13.5x = 54 \][/tex]
So,
[tex]\[ x = \frac{54}{13.5} \][/tex]
[tex]\[ x = 4 \][/tex]

Now substitute \(x = 4\) back into \(y = 4 - 1.5x\):
[tex]\[ y = 4 - 1.5(4) \][/tex]
[tex]\[ y = 4 - 6 \][/tex]
[tex]\[ y = -2 \][/tex]

Thus, the correct solution should be \((x, y) = (4, -2)\).

### Conclusion

1. Lin’s solution \((-36, 58)\) does not meet all the criteria as it does not satisfy both equations.
2. The corrected solution for the system of linear equations is \((4, -2)\).

To further verify the solution, we could graph these two equations and observe their intersection point, which should confirm our corrected solution [tex]\((4, -2)\)[/tex].
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