Question

two systems of equations are shown:4x+y=−1 x+y=2 and8x+2y=−2 −4x+cy=−8 What would the value of c need to be to get the same solution for both systems?(1 point)Responses4422-4-411-2

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.

The first system is:
1. \( 4x + y = -1 \)
2. \( x + y = 2 \)

We can solve for \( y \) in terms of \( x \) using the second equation:
\[ y = 2 - x. \]

Next, we substitute this expression for \( y \) into the first equation:
\[ 4x + (2 - x) = -1 \]
\[ 4x + 2 - x = -1 \]
Combine like terms:
\[ 3x + 2 = -1 \]
Subtract 2 from both sides:
\[ 3x = -3 \]
Divide by 3:
\[ x = -1. \]

Now we can substitute \( x = -1 \) back into the equation for \( y \):
\[ y = 2 - (-1) = 2 + 1 = 3. \]

So the solution to the first system is \( (x, y) = (-1, 3) \).

Now we will use the second system of equations to find the value of \( c \) that will yield the same solution \( (x, y) = (-1, 3) \).

The second system is:
1. \( 8x + 2y = -2 \)
2. \( -4x + cy = -8 \)

First, we substitute \( x = -1 \) and \( y = 3 \) into the first equation of the second system:
\[ 8(-1) + 2(3) = -2 \]
\[ -8 + 6 = -2 \]
This is true, so the first equation in the second system has the same solution.

Next, we need to substitute \( x = -1 \) and \( y = 3 \) into the second equation to find \( c \):
\[ -4(-1) + c(3) = -8 \]
\[ 4 + 3c = -8 \]
Subtract 4 from both sides:
\[ 3c = -12 \]
Divide by 3:
\[ c = -4. \]

Thus, the value of \( c \) needed to get the same solution for both systems is \( -4 \).

So the answer is:
\[-4\]