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?
A) 2
B) -4
C) 1
D) -2
E) 4

Answer 1

To find the same solution for both systems, we need to find the values of x and y that satisfy both equations in both systems.

For the first system, we can solve it by using the method of substitution:
x + y = 2 --> y = 2 - x
4x + y = -1 --> 4x + (2 - x) = -1 --> 3x + 2 = -1 --> 3x = -3 --> x = -1
Substituting this value of x in the first equation, we get:
y = 2 - (-1) = 3

So, the solution for the first system is x = -1 and y = 3.

Now, let's solve the second system of equations:
8x + 2y = -2 --> 4x + y = -1 * (dividing the equation by 2)
-4x + cy = -8 --> -4x + 3y = -8 (substituting c=3)

To solve this system, we can add the two equations:
(4x + y) + (-4x + 3y) = (-1) + (-8)
4x + y - 4x + 3y = -9
4y + 9y = -9
13y = -9
y = -9/13

Now, substituting the value of y back into one of the equations, we can solve for x:
4x + y = -1
4x + (-9/13) = -1
4x = -1 + 9/13
4x = -13/13 + 9/13
4x = -4/13
x = -1/13

So, the solution for the second system is x = -1/13 and y = -9/13.

Since the solution in the first system is (x, y) = (-1, 3) and the solution in the second system is (x, y) = (-1/13, -9/13), the values of c that would give the same solution in both systems is c = -4.

Therefore, the answer is B) -4.