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?
A) 2
B) -4
C) 1
D) -2
E) 4
To find the same solution for both systems, we need to find the values of x and y that satisfy both equations in both systems.
For the first system, we can solve it by using the method of substitution:
x + y = 2 --> y = 2 - x
4x + y = -1 --> 4x + (2 - x) = -1 --> 3x + 2 = -1 --> 3x = -3 --> x = -1
Substituting this value of x in the first equation, we get:
y = 2 - (-1) = 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 --> 4x + y = -1 * (dividing the equation by 2)
-4x + cy = -8 --> -4x + 3y = -8 (substituting c=3)
To solve this system, we can add the two equations:
(4x + y) + (-4x + 3y) = (-1) + (-8)
4x + y - 4x + 3y = -9
4y + 9y = -9
13y = -9
y = -9/13
Now, substituting the value of y back into one of the equations, we can solve for x:
4x + y = -1
4x + (-9/13) = -1
4x = -1 + 9/13
4x = -13/13 + 9/13
4x = -4/13
x = -1/13
So, the solution for the second system is x = -1/13 and y = -9/13.
Since the solution in the first system is (x, y) = (-1, 3) and the solution in the second system is (x, y) = (-1/13, -9/13), the values of c that would give the same solution in both systems is c = -4.
Therefore, the answer is B) -4.
To find the value of c that would give the same solution for both systems of equations, we need to solve each system separately and compare their solutions.
For the first system:--
We can solve this system using the method of substitution or elimination. For simplicity, let's use the method of elimination.
Multiply the second equation by 4 to make the coefficients of x the same in both equations:
4(4x + y) = 4(-1)
16x + 4y = -4
Now, compare both systems:
System 1:
4x + y = -1
16x + 4y = -4
From the first equation of System 1, we can multiply it by 4 to make the coefficients of y the same in both equations:
4(4x + y) = 4(-1)
16x + 4y = -4
As we can see, the second equation of System 1 is the same as the second equation of System 2. Therefore, c needs to be the same as 4 in order for the two systems to have the same solution.
So, the answer is:
E) 4
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, let's solve the first system of equations.
System 1:
4x + y = -1 (equation 1)
x + y = 2 (equation 2)
To solve this system, we can use the method of substitution. Solve equation 2 for x:
x = 2 - y
Substitute this expression for x into equation 1:
4(2 - y) + y = -1
8 - 4y + y = -1
7 - 3y = -1
-3y = -8
y = 8/3
Now substitute this value of y back into equation 2 to find x:
x + 8/3 = 2
x = 2 - 8/3
x = 6/3 - 8/3
x = -2/3
So, the solution to system 1 is x = -2/3 and y = 8/3.
Now let's solve the second system of equations.
System 2:
8x + 2y = -2 (equation 3)
-4x + cy = -8 (equation 4)
To find the value of c that gives the same solution, we need to find values of x and y that satisfy both equations 3 and 4. Notice that equation 4 is directly proportional to equation 1 from the first system. This means that we can derive the equation 4 from equation 1 by multiplying equation 1 by some constant.
Since equation 1 is 4x + y = -1, multiplying it by 2 gives us:
8x + 2y = -2 (equation 1 multiplied by 2)
Comparing equation 3 and equation 1 multiplied by 2, we can see that they are the same equation. This means that for both systems to have the same solution, c must be equal to 2.
Therefore, the value of c that would give the same solution for both systems is c = 2.
The correct answer is A) 2.