Question

unit 2 l13 practice problems solving soe by substitution date block 1 identify a solution to this system of equations: \

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

a: (-5,2) b: (-2,5) c: (-3,4) 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: $6x - 5(4 - 1.5x) = 34$ $6x - 20 - 7.5x = 34$ $-1.5x = 54$ $x = -36$ 1. check to see if lin’s solution of $(-36, 58)$ makes both equations in the system true. 2. 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:

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Problem 1

Step1: Test option A (-5,2)

Substitute $x=-5, y=2$ into:

1. $-4x + 3y = -4(-5)+3(2)=20+6=26

eq 23$

1. $x-y=-5-2=-7$

Only one equation holds, so invalid.

Step2: Test option B (-2,5)

Substitute $x=-2, y=5$ into:

1. $-4x + 3y = -4(-2)+3(5)=8+15=23$
2. $x-y=-2-5=-7$

Both equations hold.

Step3: Verify other options (optional)

Option C (-3,4):
$-4(-3)+3(4)=12+12=24
eq23$
Option D (4,-3):
$-4(4)+3(-3)=-16-9=-25
eq23$

Step1: Check Lin's substitution error

Lin substituted $y=4-1.5x$ into $6x-5y=34$, but she should have used the first equation to substitute into the second, or vice versa. She mixed up the equations: $y=4-1.5x$ comes from rearranging $3x+2y=8$, so it should be substituted into $6x-5y=34$, but her expansion is wrong:
Correct expansion:
$6x - 5(4-1.5x) = 34$
$6x -20 +7.5x = 34$ (Lin wrote $-7.5x$ instead of $+7.5x$)

Step2: Solve for x correctly

Combine like terms:
$6x+7.5x=34+20$
$13.5x=54$
$x=\frac{54}{13.5}=4$

Step3: Solve for y

Substitute $x=4$ into $y=4-1.5x$:
$y=4-1.5(4)=4-6=-2$

Step4: Verify solution (4,-2)

Check in $6x-5y=34$:
$6(4)-5(-2)=24+10=34$
Check in $3x+2y=8$:
$3(4)+2(-2)=12-4=8$
Both equations hold. Lin's mistake was a sign error when distributing $-5$ to $-1.5x$.

Step5: Check Lin's solution (-36,58)

Substitute $x=-36,y=58$ into $3x+2y=8$:
$3(-36)+2(58)=-108+116=8$ (holds)
Substitute into $6x-5y=34$:
$6(-36)-5(58)=-216-290=-506
eq34$ (fails)

Answer:

B: (-2, 5)

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Problem 2