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
-4
-4
1
1
4
4
2
2
-2
To find the value of c that would give the same solution for both systems of equations, we need to solve the systems and compare the values of y in each system.
For the first system:
4x + y = -1
x + y = 2
Rearranging the second equation, we get:
y = 2 - x
Substituting this value into the first equation, we have:
4x + (2 - x) = -1
Simplifying, we get:
3x + 2 = -1
3x = -3
x = -1
Substituting x = -1 into the second equation, we have:
y = 2 - (-1)
y = 2 + 1
y = 3
Therefore, the solution for the first system is (-1, 3).
For the second system:
8x + 2y = -2
-4x + cy = -8
We need to find the value of c that would make the y-values in both systems equal. From the first system, we know that y = 3. Substituting y = 3 into the second equation, we have:
-4x + 3c = -8
Simplifying, we get:
4x - 3c = 8
Comparing this equation with the first equation in the second system, which is 8x + 2y = -2, we can see that the coefficient of y in both equations must be the same. Therefore, we need to find the value of c that would make the coefficient of y in the second equation equal to 2.
From the equation 4x - 3c = 8, we can see that -3c = 2 when c = -4.
Therefore, the value of c that would give the same solution for both systems is -4.
its -4 I think
To find the value of c that would give the same solution for both systems, we can compare the coefficients of y in both systems.
In the first system:
4x + y = -1
In the second system:
-4x + cy = -8
For both systems to have the same solution, the coefficients of y should be the same. So we can set up the equation:
1 = cy
To solve for c, we can rearrange the equation:
c = 1/y
Since there is no given value for y in the question, we cannot determine the specific value of c that would give the same solution for both systems. Therefore, none of the provided options (-4, 1, 4, 2, -2) is correct.
To find the value of c that will result in the same solution for both systems of equations, we need to solve each system and compare the solutions.
For the first system:
1. Rewrite the equations:
4x + y = -1 => y = -4x - 1
x + y = 2 => y = -x + 2
2. Set the right sides of the two equations equal to each other to find a common solution:
-4x - 1 = -x + 2
3. Solve for x:
-4x + x = 2 + 1
-3x = 3
x = -1
4. Substitute the value of x back into one of the original equations to solve for y:
x + y = 2
-1 + y = 2
y = 3
So the solution to the first system is x = -1 and y = 3.
Now let's solve the second system with the unknown value c:
1. Rewrite the equations:
8x + 2y = -2 => y = -4x - 1/2
-4x + cy = -8 => cy = 4x - 8
2. Substitute the value of y from the first equation into the second equation:
c*(-4x - 1/2) = 4x - 8
3. Simplify the equation:
-4cx - c/2 = 4x - 8
4. Rearrange the equation:
4cx + 4x = -c/2 + 8
5. Combine like terms:
(4c + 4)x = -c/2 + 8
For the two systems to have the same solution, the values of x and y must be the same. This means that the coefficients of x and the constants on the right side of the equations must also be the same.
Comparing the equations:
-4x - 1 = 4cx + 4x
-1 = 4cx + 8x
Setting the coefficients of x equal:
4c + 8 = 0
Solving for c:
4c = -8
c = -8/4
c = -2
Therefore, the value of c that would result in the same solution for both systems is -2.