To determine the value of \( c \) that would make the fourth equation consistent with the solutions of the other equations, we need to analyze the given systems of equations.
First, let's rewrite the first system of equations:
- \( 4x + t = -1 \)
- \( x + y = 2 \)
- \( 8x + 2y = -2 \)
- \( -4x + cy = -8 \)
We start with the second and third equations, as they contain both \( x \) and \( y \).
From equation 2: \[ y = 2 - x \]
Now substitute \( y \) from equation 2 into equation 3: \[ 8x + 2(2 - x) = -2 \] Expanding this: \[ 8x + 4 - 2x = -2 \] Combine like terms: \[ 6x + 4 = -2 \] Subtract 4 from both sides: \[ 6x = -6 \] Divide by 6: \[ x = -1 \]
Now substitute \( x = -1 \) back into equation 2 to find \( y \): \[ y = 2 - (-1) = 2 + 1 = 3 \]
Thus, the solution for the values of \( x \) and \( y \) is \( x = -1 \) and \( y = 3 \).
Next, we need to ensure that the values \( x = -1 \) and \( y = 3 \) satisfy the fourth equation \( -4x + cy = -8 \): \[ -4(-1) + c(3) = -8 \] Simplifying: \[ 4 + 3c = -8 \] Subtract 4 from both sides: \[ 3c = -12 \] Divide by 3: \[ c = -4 \]
Thus, for the fourth equation to have the same solution as the other equations, \( c \) must be \( -4 \).
The final answer is: \[ \boxed{-4} \]