Question

Question
Two systems of equations are shown:

4x+y=−1


x+y=2


and

8x+2y=−2


−4x+cy=−8


What would the value of c need to be to get the same solution for both systems?

(1 point)
Responses

-2
-2

4
4

-4
-4

2
2

1
1
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Answer 1

To determine the value of \( c \) that will yield the same solution for both systems of equations, we first need to analyze the solutions of the first system:

1. **First System of Equations:**
\[
4x + y = -1 \quad \text{(1)}
\]
\[
x + y = 2 \quad \text{(2)}
\]

From equation (2), we can express \( y \) in terms of \( x \):
\[
y = 2 - x
\]

Now, we substitute this expression for \( y \) into equation (1):
\[
4x + (2 - x) = -1
\]
This simplifies to:
\[
4x + 2 - x = -1
\]
\[
3x + 2 = -1
\]
\[
3x = -3
\]
\[
x = -1
\]

Now we substitute \( x = -1 \) back into equation (2) to find \( y \):
\[
-1 + y = 2
\]
\[
y = 3
\]

Thus, the solution for the first system is \( x = -1 \) and \( y = 3 \).

2. **Second System of Equations:**
\[
8x + 2y = -2 \quad \text{(3)}
\]
\[
-4x + cy = -8 \quad \text{(4)}
\]

Let's check if the first equation of this system has the same solution \( (x, y) = (-1, 3) \).

Substituting \( x = -1 \) and \( y = 3 \) into equation (3):
\[
8(-1) + 2(3) = -2
\]
\[
-8 + 6 = -2
\]
This is valid.

Now for equation (4):
\[
-4(-1) + c(3) = -8
\]
\[
4 + 3c = -8
\]
Solving for \( c \):
\[
3c = -8 - 4
\]
\[
3c = -12
\]
\[
c = -4
\]

Thus, the value of \( c \) that would yield the same solution for both systems of equations is \(-4\).

The answer is **-4**.