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
-4
-4
1
1
2
2
-2
In order for the two systems of equations to have the same solution, the coefficients of the corresponding variables in each equation must be proportional.
Comparing the first equation of the first system (4x + y = -1) to the first equation of the second system (8x + 2y = -2), we can see that the coefficient of x in the first equation is half of the coefficient of x in the second equation (4x vs 8x).
Similarly, the coefficient of y in the first equation is also half of the coefficient of y in the second equation (1y vs 2y).
Therefore, the value of c needs to be 2 in order for both systems to have the same solution.
To find the value of c that would give the same solution for both systems of equations, we need to manipulate the second system of equations to match the coefficients of the first system.
Let's start by looking at the first equation of each system:
System 1: 4x + y = -1
System 2: 8x + 2y = -2
We can see that the coefficient of x in the second equation is twice as large as the coefficient of x in the first equation. To match the coefficients, we need to divide the second equation by 2:
(8x + 2y)/2 = -2/2
4x + y = -1
Now, let's look at the second equation of each system:
System 1: x + y = 2
System 2: -4x + cy = -8
To match the coefficients, we need to multiply System 2 by -1:
-1*(-4x) + (-1)*cy = -1*(-8)
4x - cy = 8
Now we have:
4x + y = -1
4x - cy = 8
From these two equations, we can see that the coefficient of y in System 1 is 1, while the coefficient of y in System 2 is -c. In order to get the same solution for both systems, we need these coefficients to be the same. Therefore, we set -c = 1:
-c = 1
To solve for c, we can multiply both sides by -1:
c = -1
Therefore, the value of c that would give the same solution for both systems of equations is -1. So the correct answer is:
-2
To find the value of c that would give the same solution for both systems of equations, we can compare the slopes of the two systems. If the slopes are equal, then c would be the value that makes the slopes equal.
By rearranging the first equation of the second system, we have:
-4x + cy = -8
Comparing the two equations, we can see that the slope of the first equation in the first system is 4, while the slope of the first equation in the second system is -4. To make the slopes equal, c would need to be 4.
Therefore, the correct answer is:
c = 4