Two systems of equations are shown:%0D%0A%0D%0A4x+y=−1%0D%0A4%0D%0A%0D%0A+%0D%0A%0D%0A=%0D%0A−%0D%0A1%0D%0A %0D%0Ax+y=2%0D%0A%0D%0A+%0D%0A%0D%0A=%0D%0A2%0D%0A %0D%0Aand%0D%0A%0D%0A8x+2y=−2%0D%0A8%0D%0A%0D%0A+%0D%0A2%0D%0A%0D%0A=%0D%0A−%0D%0A2%0D%0A %0D%0A−4x+cy=−8%0D%0A−%0D%0A4%0D%0A%0D%0A+%0D%0A%0D%0A%0D%0A=%0D%0A−%0D%0A8%0D%0A %0D%0AWhat would the value of c need to be to get the same solution for both systems?%0D%0A%0D%0A(1 point)%0D%0AResponses%0D%0A%0D%0A1%0D%0A1%0D%0A%0D%0A-2%0D%0A-2%0D%0A%0D%0A-4%0D%0A-4%0D%0A%0D%0A2%0D%0A2%0D%0A%0D%0A4
To find the value of c that would give the same solution for both systems, we can compare the coefficients of x and y in each equation.
For the system 4x + y = -1 and x + y = 2:
- the coefficient of x is 4
- the coefficient of y is 1
For the system 8x + 2y = -2 and -4x + cy = -8:
- the coefficient of x is 8 in the first equation and -4 in the second equation
- the coefficient of y is 2 in the first equation and c in the second equation
To get the same solution for both systems, the coefficients of x and y in the second system should be twice the coefficients in the first system.
Therefore, c should be equal to 2.
So the value of c that would give the same solution for both systems is 2.
To get the same solution for both systems of equations, the value of c needs to be -4.
Here's the step-by-step solution for each system of equations:
1) System of Equations:
4x + y = -1
x + y = 2
To find the solution, let's solve the second equation for y and substitute it into the first equation:
x + y = 2 [Equation 1]
y = 2 - x
Substituting y = 2 - x into the first equation:
4x + (2 - x) = -1
4x + 2 - x = -1
3x + 2 = -1
3x = -1 - 2
3x = -3
x = -1
Substituting x = -1 into the second equation:
-1 + y = 2
y = 2 + 1
y = 3
So, the solution to the first system of equations is x = -1 and y = 3.
2) System of Equations:
8x + 2y = -2
-4x + cy = -8
We need to find the value of c such that the solution matches the previous system.
Substituting x = -1 and y = 3 into the first equation:
8(-1) + 2(3) = -2
-8 + 6 = -2
-2 = -2
Now, let's substitute the same values into the second equation:
-4(-1) + c(3) = -8
4 + 3c = -8
3c = -8 - 4
3c = -12
c = -12/3
c = -4
Therefore, the value of c needs to be -4 to get the same solution for both systems.
To find the value of c that would make both systems have the same solution, we can start by solving each system individually.
First, let's solve the first system of equations:
4x + y = -1 ...(Equation 1)
x + y = 2 ...(Equation 2)
To solve this system, we can use the method of elimination.
Multiply Equation 2 by 4 so that the coefficients of the x-terms will cancel out when we add the equations:
4(x + y) = 4(2)
4x + 4y = 8 ...(Equation 3)
Now, subtract Equation 1 from Equation 3 to eliminate the x-term:
(4x + 4y) - (4x + y) = 8 - (-1)
3y = 9
y = 3
Substitute the value of y back into Equation 2 to find x:
x + 3 = 2
x = -1
Therefore, the solution to the first system is x = -1 and y = 3.
Now let's solve the second system of equations:
8x + 2y = -2 ...(Equation 4)
-4x + cy = -8 ...(Equation 5)
Again, we'll use the elimination method to solve this system.
Multiply Equation 4 by 2 and Equation 5 by 4 so that the coefficients of the x-terms will cancel out.
16x + 4y = -4 ...(Equation 6)
-16x + 4cy = -32 ...(Equation 7)
Add Equation 6 to Equation 7 to eliminate the x-term:
(16x + 4y) + (-16x + 4cy) = -4 + (-32)
4y + 4cy = -36
Factor out 4 from the left side of the equation:
4(y + cy) = -36
(y + cy) = -9
Divide both sides of the equation by (y + cy):
1 = -9 / (y + cy)
Since both systems have the same solution, the numerators and denominators on both sides of the equation must be equal.
Comparing the equation 1 = -9 / (y + cy) to the equation 1 = -9 / 3 (from the first system), we can see that the denominator in the second system is equal to 3. Therefore, in order for both systems to have the same solution, c must equal 3.
Hence, the value of c that would make both systems have the same solution is c = 3.