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
2
2
-4
-4
4
4
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To get the same solution for both systems, the two systems must be equivalent. To determine the value of c that would make the two systems equivalent, we can compare the equations in each system.
In the first system, we have:
4x + y = -1
x + y = 2
In the second system, we have:
8x + 2y = -2
-4x + cy = -8
We can see that the second equation in the second system is equivalent to the second equation in the first system. Thus, the value of c that would make the two systems equivalent is the coefficient of y from the second equation in the second system, which is -2.
Therefore, the answer is c = -2.
To find the value of c that would give the same solution for both systems, we need to solve both systems of equations and compare the solutions.
First, let's solve the first system:
4x + y = -1 (equation 1)
x + y = 2 (equation 2)
We can use the method of substitution to solve these equations. Solve equation 2 for x:
x = 2 - y
Now substitute this value of x in equation 1:
4(2 - y) + y = -1
8 - 4y + y = -1
7 - 3y = -1
-3y = -8
y = 8/3
Substitute this value of y back into equation 2:
x + (8/3) = 2
x = 2 - (8/3)
x = 6/3 - 8/3
x = -2/3
So the solution for the first system is x = -2/3 and y = 8/3.
Now let's solve the second system:
8x + 2y = -2 (equation 3)
-4x + cy = -8 (equation 4)
We need to find the value of c that would make the solution of this system the same as the first system (x = -2/3 and y = 8/3).
Substitute the values of x and y into equation 3:
8(-2/3) + 2(8/3) = -2
-16/3 + 16/3 = -2
0 = -2
Since this equation is not true for any value of c, there is no value of c that would give the same solution for both systems of equations. Therefore, the answer is none of the provided options.
To find the value of c that would give the same solution for both systems of equations, we need to solve the first system and then substitute the values of x and y into the second system.
Let's solve the first system of equations:
4x + y = -1 (1)
x + y = 2 (2)
We can solve this system using the method of elimination. Subtracting equation (2) from equation (1), we get:
(4x + y) - (x + y) = -1 - 2
3x = -3
x = -1
Substituting x = -1 into equation (2), we get:
-1 + y = 2
y = 3
So, the solution to the first system is x = -1 and y = 3.
Now, let's substitute these values into the second system of equations:
8x + 2y = -2 (3)
-4x + cy = -8 (4)
Substituting x = -1 and y = 3 into equation (3):
8(-1) + 2(3) = -2
-8 + 6 = -2
-2 = -2
Substituting x = -1 and y = 3 into equation (4):
-4(-1) + c(3) = -8
4 + 3c = -8
3c = -12
c = -4
So, in order to get the same solution for both systems, the value of c needs to be -4. Therefore, the correct answer is -4.