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].
