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
1
1
-2
-2
-4
-4
4
4
2
To get the same solution for both systems, the two systems must be equivalent. This means that the two equations in the second system must be multiples of the corresponding equations in the first system.
Comparing the equations in the first system, we can see that multiplying the second equation by 2 will give us the second equation in the second system:
2(x+y) = 2(2)
2x+2y = 4
Therefore, the value of c needs to be 2.
To get the same solution for both systems of equations, we can compare the slopes of the equations.
For the first system:
4x + y = -1
The slope is -4.
x + y = 2
The slope is -1.
For the second system:
8x + 2y = -2
The slope is -4.
-4x + cy = -8
The slope is -4c.
To make the slopes equal, we need -4c = -4.
Dividing both sides of the equation by -4, we get c = 1.
Therefore, the value of c that would make the two systems have the same solution is 1.
To find the value of c that would give the same solution for both systems of equations, we can compare the two systems and look for a pattern.
First, 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 substitution. From equation (2), we can express y in terms of x:
y = 2 - x
Substituting this value of y into equation (1):
4x + (2 - x) = -1
Simplifying:
3x + 2 = -1
3x = -3
x = -1
Now, substitute the value of x into equation (2):
(-1) + y = 2
y = 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 ...(3)
-4x + cy = -8 ...(4)
We don't know the value of c yet, so we'll keep it as a variable for now.
Let's solve equation (4) for y in terms of x:
-4x + cy = -8
cy = -8 + 4x
y = (-8 + 4x)/c ...(5)
Substituting equation (5) into equation (3) and simplifying:
8x + 2((-8 + 4x)/c) = -2
8x + (-16 + 8x)/c = -2
Multiplying both sides by c to remove the denominator:
8cx + (-16 + 8x) = -2c
8cx - 16 + 8x = -2c
16x + 8cx - 16 = -2c
16x + (8c - 2c)x = 16
16x + 6cx = 16
(16 + 6c)x = 16
x = 16 / (16 + 6c)
To get the same solution for both systems, the value of x should be the same for both systems. Therefore, x = -1 as we found earlier.
Substituting x = -1 into the equation above:
-1 = 16 / (16 + 6c)
To find the value of c, we need to solve this equation for c.
Multiply both sides by (16 + 6c):
-1(16 + 6c) = 16
Expanding and simplifying:
-16 - 6c = 16
Add 16 to both sides:
-6c = 32
Divide both sides by -6 to solve for c:
c = 32 / -6
c = -16/3
Therefore, the value of c that would give the same solution for both systems is c = -16/3.