Question

How many solutions does the system of equations have?

y=−2x+12

4y+x−20=0

Responses

Two solutions

One solution

No solution

Infinitely many solutions

Answer 1

[AI-generated answer — Bot GPT 3.5]

One solution

Answer 2

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

To determine the number of solutions the system of equations has, we can solve the system by either substitution or elimination.

First, let's solve by substitution. We'll start by solving the first equation for y:

y = -2x + 12

Next, we'll substitute this expression for y in the second equation:

4(-2x + 12) + x - 20 = 0

Simplifying, we get:

-8x + 48 + x - 20 = 0

Combining like terms:

-7x + 28 = 0

Next, we'll isolate x:

-7x = -28

Dividing both sides by -7:

x = 4

Now that we have the value of x, we can substitute it back into one of the equations to find y. Let's use the first equation:

y = -2(4) + 12
y = -8 + 12
y = 4

Therefore, the system has one solution: x = 4 and y = 4.

So, the correct response is:

One solution

Answer 3

[AI-generated answer — Explain Bot]

To determine the number of solutions for the system of equations, we can use the method of substitution or elimination. Let's use the elimination method in this case.

The given system of equations is:

y = -2x + 12 ---(1)

4y + x - 20 = 0 ---(2)

To use the elimination method, we need to multiply one or both equations by appropriate constants such that the coefficients of one variable will cancel out when we add or subtract the equations.

To eliminate the x-term, we can multiply equation (1) by 4, so it becomes:

4y = -8x + 48

Now we have:

4y + x - 20 = 0 ---(2)
4y = -8x + 48 ---(3)

We can subtract equation (3) from equation (2) to eliminate the y-term:

4y + x - 20 - (4y) = 0 - (-8x + 48)
x - 20 = 8x - 48

Next, solve for x:

7x = 28
x = 4

Substitute the value of x into equation (1) to solve for y:

y = -2x + 12
y = -2(4) + 12
y = -8 + 12
y = 4

Therefore, the system of equations has one solution, which is x = 4 and y = 4.