Question: 2. Lin is solving this system of equations:
{6x - 5y = 34
3x + 2y = 8}
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:
6x - 5(4 - 1.5x) = 34
6x - 20 - 7.5x = 34
-1.5x = 54
x = -36
y = 4 - 1.5x
y = 4 - 1.5 \cdot (-36)
y = 58
a. Check to see if Lin's solution of (-36,58)
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.
1. **State the problem:**
We are given the system of equations:
$$6x - 5y = 34$$
$$3x + 2y = 8$$
Lin found a solution $(-36, 58)$ by isolating $y$ in the second equation and substituting into the first.
2. **Check Lin's solution in the first equation:**
Substitute $x = -36$ and $y = 58$ into $$6x - 5y = 34$$
Calculate:
$$6(-36) - 5(58) = -216 - 290 = -506$$
Since $$-506 \neq 34$$, the solution does not satisfy the first equation.
3. **Check Lin's solution in the second equation:**
Substitute $x = -36$ and $y = 58$ into $$3x + 2y = 8$$
Calculate:
$$3(-36) + 2(58) = -108 + 116 = 8$$
This satisfies the second equation.
4. **Conclusion:**
Lin's solution $(-36, 58)$ satisfies the second equation but not the first.
5. **Find and explain the mistake:**
Lin's mistake occurred when simplifying the first equation after substitution:
Starting from:
$$6x - 5(4 - 1.5x) = 34$$
Distribute:
$$6x - 20 + 7.5x = 34$$
Combine like terms:
$$13.5x - 20 = 34$$
Add 20 to both sides:
$$13.5x = 54$$
Divide both sides by 13.5:
$$x = \frac{54}{13.5}$$
Simplify:
$$x = 4$$
6. **Find $y$ using $x=4$:**
Use the expression for $y$:
$$y = 4 - 1.5x = 4 - 1.5 \times 4 = 4 - 6 = -2$$
7. **Verify the corrected solution $(4, -2)$:**
Check first equation:
$$6(4) - 5(-2) = 24 + 10 = 34$$
Check second equation:
$$3(4) + 2(-2) = 12 - 4 = 8$$
Both equations are satisfied.
**Final answer:**
Lin's solution $(-36, 58)$ is incorrect due to a sign error in distributing the $-5$.
The correct solution is $$\boxed{(4, -2)}$$.