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 1 1 -4 -4 2 2 4

Answer 1

To find the value of \( c \) such that both systems of equations have the same solution, we first need to determine the solution to the first system of equations:

1. \( 4x + y = -1 \) (Equation 1)
2. \( x + y = 2 \) (Equation 2)

To solve this system, we can use the substitution or elimination method. Let's use substitution.

From Equation 2, we isolate \( y \):

\[
y = 2 - x
\]

Now, substitute \( y \) in Equation 1:

\[
4x + (2 - x) = -1
\]

This simplifies to:

\[
4x + 2 - x = -1
\]

Combining like terms, we get:

\[
3x + 2 = -1
\]

Subtracting 2 from both sides:

\[
3x = -3
\]

Dividing both sides by 3:

\[
x = -1
\]

Now that we have \( x \), we can find \( y \) by substituting \( x = -1 \) back into Equation 2:

\[
y = 2 - (-1) = 2 + 1 = 3
\]

So the solution to the first system is:

\[
(x, y) = (-1, 3)
\]

Now we will explore the second system:

1. \( 8x + 2y = -2 \) (Equation 3)
2. \( -4x + cy = -8 \) (Equation 4)

Notice that Equation 3 can be simplified. Dividing the entire equation by 2 gives us:

\[
4x + y = -1
\]

This is the same as Equation 1 from the first system, which we used to find that:

\[
y = -1 - 4x
\]

For the second equation (Equation 4) to have the same solution, we already found that \( x = -1 \) and \( y = 3 \). Let's substitute these values into Equation 4:

\[
-4(-1) + c(3) = -8
\]

This simplifies to:

\[
4 + 3c = -8
\]

Subtracting 4 from both sides gives:

\[
3c = -12
\]

Now, divide by 3:

\[
c = -4
\]

Therefore, the necessary value for \( c \) to obtain the same solution as the first system is:

\[
\boxed{-4}
\]