Question

1. identify a solution to this system of equations: \\(\

$$\begin{cases}-4x + 3y=23\\x - y = 7\\end{cases}$$

\\) a) (-8, -2) b) (-2, 5) c) (8, 4) d) (4, -3) 2. lin is solving this system of equations: \\(\

$$\begin{cases}6x - 5y=34\\3x + 2y = 8\\end{cases}$$

\\) 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 = 4-1.5cdot(-36)) (y = 58) a. check to see if lins solution of ((-36,58)) makes both equations in the system true. b. if your answer to the previous question is
o,\ find and explain her mistake. if your answer is \yes,\ graph the equations to verify the solution of the system.

Explanation:

Step1: Substitute into first - equation

Substitute \(x=-36\) and \(y = 58\) into \(6x-5y=34\).
\[6\times(-36)-5\times58= - 216-290=-506
eq34\]

Step2: Substitute into second - equation

Substitute \(x = - 36\) and \(y = 58\) into \(3x + 2y=8\).
\[3\times(-36)+2\times58=-108 + 116 = 8\]
Since the point \((-36,58)\) does not satisfy the first equation \(6x-5y = 34\), Lin's solution is incorrect.

Step3: Find the mistake

When Lin distributed \(-5\) in \(6x-5(4 - 1.5x)=34\), she made a sign error.
The correct distribution is \(6x-20 + 7.5x=34\), not \(6x-20-7.5x=34\).
Combining like - terms gives \(13.5x=54\), then \(x = 4\).
Substitute \(x = 4\) into \(y=4-1.5x\), we get \(y=4-1.5\times4=4 - 6=-2\).

Answer:

a. No.
b. Lin made a sign error when distributing \(-5\) in \(6x-5(4 - 1.5x)=34\). The correct distribution should be \(6x-20 + 7.5x=34\), not \(6x-20-7.5x=34\). The correct solution of the system is \((4,-2)\).