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 -2 -2 -4
To get the same solution for both systems, the two systems must have the same slopes and the same y-intercepts.
For the first system, we can rewrite the second equation as y = 2 - x.
Comparing with the first equation, we can see that the slope is -1 for both equations and the y-intercept is 2 for both equations.
For the second system, we can rewrite the second equation as y = cx - 2.
Comparing with the first equation, we can see that the slope is c for both equations and the y-intercept is -2 for both equations.
Therefore, to get the same solution for both systems, the value of c must be equal to -1.
Answer: -1
To find the value of c that would make both systems have the same solution, we need to compare the two systems and determine the relationship between the coefficients of the variables.
First, let's analyze the coefficient of x in each equation of both systems:
In the first system, the coefficient of x in the first equation is 4, and in the second equation, it is 1.
In the second system, the coefficient of x in the first equation is 8, and in the second equation, it is -4.
Since we want both systems to have the same solution, the coefficients of x in both systems must be equal. Therefore, we set up the equation:
4 = 8c
By simplifying this equation, we can solve for the value of c:
4/8 = c
1/2 = c
So, the value of c that would make both systems have the same solution is c = 1/2.
Therefore, the correct answer is 2.
To find the value of c that makes both systems have the same solution, we need to equate the two systems and solve for c.
The first system can be written as:
4x + y = -1 ...(1)
x + y = 2 ...(2)
The second system can be written as:
8x + 2y = -2 ...(3)
-4x + cy = -8 ...(4)
To equate the systems, we can solve for y in terms of x from equations (2) and (4) to eliminate y:
From equation (2):
y = 2 - x
From equation (4):
cy = -8 + 4x
y = (-8 + 4x)/c
Equating the two expressions for y:
2 - x = (-8 + 4x)/c ...(5)
To find c, we need to solve equation (5) for x. Let's multiply both sides by c to eliminate the fraction:
2c - cx = -8 + 4x
cx + 4x = 2c + 8
x(c + 4) = 2c + 8
x = (2c + 8) / (c + 4)
Now we substitute this value of x into equation (2) to find y:
x + y = 2
(2c + 8) / (c + 4) + y = 2
(2c + 8) + y(c + 4) = 2(c + 4)
(2c + 8) + y(c + 4) = 2c + 8
(2c + 8) + (y(c + 4) - 2c - 8) = 0
(2c + 8) + (cy + 4y - 2c - 8) = 0
cy + 4y = 0
Since this equation needs to hold true for any value of y, the coefficient of y must be zero:
c + 4 = 0
c = -4
Therefore, the value of c that makes both systems have the same solution is -4.