Two systems of equations are shown:
4x+y=−1
x+y=2
8x+2y=−2
−4x+cy=−8
What would the value of c in the second system need to be to get the same solution for both systems?
To find the value of c that will give the same solution for both systems, we can compare the equations of the two systems.
In the first system, we have:
4x + y = -1 ...(Equation 1)
x + y = 2 ...(Equation 2)
In the second system, we have:
8x + 2y = -2 ...(Equation 3)
-4x + cy = -8 ...(Equation 4)
To solve for the same solution in both systems, the coefficients of x and y in the respective equations should be the same.
Comparing the coefficients in Equation 1 and Equation 3:
4x = 8x
This implies that 4 = 8, which is not true. Therefore, we cannot find a value of c that will make the two systems have the same solution.
To find the value of c in the second system of equations that would result in the same solution as the first system, we need to equate the coefficients of y in both systems.
In the first system, the coefficient of y is 1.
In the second system, the coefficient of y is c.
To have the same solution, the coefficient of y in both systems needs to be the same. Therefore, c should be equal to 1.
To find the value of c in the second system that will give the same solution as the first system, we need to solve both systems of equations and compare the solutions.
Let's solve the first system of equations:
4x + y = -1 ... (Equation 1)
x + y = 2 ... (Equation 2)
To solve this, we can use the method of substitution. Solving Equation 2 for x, we get:
x = 2 - y
Substituting this value of x in Equation 1, we have:
4(2 - y) + y = -1
8 - 4y + y = -1
8 - 3y = -1
-3y = -9
y = 3
Now, substitute the value of y back into Equation 2 to find x:
x + 3 = 2
x = -1
So, 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 3)
-4x + cy = -8 ... (Equation 4)
To find the value of c that gives the same solution, we need to solve Equation 3 and Equation 4. Since we already know the solution to the first system, we can substitute those values into Equation 3 and Equation 4.
Substituting x = -1 and y = 3 into Equation 3, we have:
8(-1) + 2(3) = -2
-8 + 6 = -2
-2 = -2
Substituting x = -1 and y = 3 into Equation 4, we have:
-4(-1) + c(3) = -8
4 + 3c = -8
3c = -12
c = -4
Therefore, the value of c in the second system needs to be -4 in order to get the same solution as the first system.