Two systems of equations are shown:

Question

4x+y=−1
x+y=2

8x+2y=−2
−4x+cy=−8

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

Answer 1

To find the value of c that would give the same solution for both systems, we need to solve each system separately and compare the solutions.

First system:
4x+y=−1
x+y=2

We can solve this system by using the method of substitution. From the second equation, we can isolate the variable x:
x = 2 - y

Now, substitute this value of x into the first equation:
4(2-y) + y = -1
8-4y + y = -1
8 - 3y = -1
-3y = -9
y = 3

Substituting the value of y back into the second equation:
x + 3 = 2
x = -1

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

Second system:
8x+2y=−2
−4x+cy=−8

We can solve this system by using the method of substitution. From the second equation, we can isolate the variable x:
-4x = -cy - 8
x = (cy + 8) / 4

Now, substitute this value of x into the first equation:
8((cy + 8) / 4) + 2y = -2
2cy + 16 + 2y = -2
2cy + 2y = -18
2y(c + 1) = -18
y = -18 / (2(c + 1))
y = -9 / (c + 1)

Substituting the value of y back into the second equation:
-4x + c(-9 / (c + 1)) = -8
-4x - 9c / (c + 1) = -8
-4x = -8 + 9c / (c + 1)
x = (-8 + 9c / (c + 1)) / -4
x = (8 - 9c / (c + 1)) / 4

So the solution for the second system is x = (8 - 9c / (c + 1)) / 4 and y = -9 / (c + 1).

For both systems to have the same solution, the values of x and y should be the same. Comparing the solutions, we can set the equations equal to each other:

-1 = (8 - 9c / (c + 1)) / 4
-1 = 8 - 9c / (c + 1) / 4
-4 = 8 - 9c / (c + 1)
-36 = - 18c / (c + 1)
36(c + 1) = 18c
36c + 36 = 18c
36c - 18c = - 36
18c = -36
c = -36 / 18
c = -2

Therefore, the value of c needed to get the same solution for both systems is c = -2.