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. Graph shape: two linear equations from the system, one line for 6x - 5y = 34 and one line for 3x + 2y = 8, intersecting near the center; position_hint center

1. **State the problem:** We are given the system of equations: $$6x - 5y = 34$$ $$3x + 2y = 8$$ Lin found a solution $(-36, 58)$ by substitution and we need to check if this solution satisfies both equations. 2. **Check the 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 the 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:** The solution $(-36, 58)$ satisfies the second equation but not the first. Therefore, Lin's solution is incorrect. 5. **Find and explain the mistake:** Lin's substitution step: $$6x - 5(4 - 1.5x) = 34$$ Expanding: $$6x - 20 + 7.5x = 34$$ Note the sign error: the minus before the parentheses applies to both terms inside, so it should be: $$6x - 20 + 7.5x = 34$$ But Lin wrote: $$6x - 20 - 7.5x = 34$$ which is incorrect. Correct expansion: $$6x - 5(4 - 1.5x) = 6x - 20 + 7.5x = 13.5x - 20$$ Set equal to 34: $$13.5x - 20 = 34$$ Add 20 to both sides: $$13.5x = 54$$ Divide both sides by 13.5: $$x = \frac{54}{13.5}$$ $$x = 4$$ 6. **Find $y$ using $x=4$:** $$y = 4 - 1.5 \cdot 4 = 4 - 6 = -2$$ 7. **Verify the corrected solution $(4, -2)$:** First equation: $$6(4) - 5(-2) = 24 + 10 = 34$$ Second equation: $$3(4) + 2(-2) = 12 - 4 = 8$$ Both equations are satisfied. 8. **Graph the equations to verify the solution:** - Equation 1: $$6x - 5y = 34$$ can be rewritten as $$y = \frac{6}{5}x - \frac{34}{5}$$ - Equation 2: $$3x + 2y = 8$$ can be rewritten as $$y = 4 - 1.5x$$ The lines intersect at $(4, -2)$, confirming the solution. **Final answer:** Lin's solution $(-36, 58)$ is incorrect due to a sign error in expanding the substitution. The correct solution is $$\boxed{(4, -2)}$$.