Question

wo 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?

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the value of c that would result in the same solution for both systems, we need to compare 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.

For the solutions to be the same, the coefficient of y in both systems must be equal.

Therefore, c must be equal to 1 for the two systems to have the same solution.

Answer 2

[AI-generated answer — Step-by-Step Bot]

To find the value of c that would make both systems of equations have the same solution, we need to set the two systems equal to each other and compare their coefficients.

For the first system:

4x + y = -1 --------(1)
x + y = 2 --------(2)

For the second system:

8x + 2y = -2 --------(3)
-4x + cy = -8 --------(4)

To compare the coefficients of the two systems, we can rearrange equation (3) by dividing both sides by 2:

4x + y = -1 --------(5)

Now we can compare equation (2) and equation (5):

x + y = 2 --------(2)
4x + y = -1 --------(5)

Comparing the coefficients of x and y in both equations, we can see that the coefficient of y is the same in both equations. Therefore, to get the same solution for both systems, the coefficient of y in equation (4) needs to be 1, just like in equation (1). Hence, the value of c would need to be 1.

Answer 3

[AI-generated answer — Explain Bot]

To find the value of c that would make both systems have the same solution, we need to compare the coefficients of the variables in the two systems.

Let's start with the first system:
4x + y = -1 -- Equation 1
x + y = 2 -- Equation 2

Now let's consider the second system:
8x + 2y = -2 -- Equation 3
-4x + cy = -8 -- Equation 4

To have the same solution for both systems, the coefficients of x and y in Equation 3 should be multiples of the coefficients of x and y in Equation 1, and the coefficients of x and y in Equation 4 should be multiples of the coefficients of x and y in Equation 2.

Comparing the coefficients, we can see that the coefficient of x in Equation 3 is 8, while the coefficient of x in Equation 1 is 4. To make them multiples, we need to double Equation 1. Similarly, the coefficient of y in Equation 3 is 2, while the coefficient of y in Equation 1 is 1. We also need to double Equation 1 to match Equation 3.

Now let's compare the coefficients in Equation 4 with those in Equation 2. The coefficient of x in Equation 4 is -4, while the coefficient of x in Equation 2 is 1. To make them multiples, we need to multiply Equation 2 by -4. The coefficient of y in Equation 4 is c, while the coefficient of y in Equation 2 is 1. So, to have the same solution, the coefficient of y in Equation 4 should also be c.

After making these adjustments, the two systems become:
8x + 2y = -2 -- Equation 3 (same as Equation 1 after doubling)
-4x + cy = -8 -- Equation 4 (same as Equation 2 after multiplying by -4)

So, to make both systems have the same solution, the value of c should be 2.