Question

How many solutions does the system of equations have?

y=−2x+12

4y+x−20=0
(1 point)
Responses

a Infinitely many solutions

b No solution

c Two solutions

d One solution

Answer 1

[AI-generated answer — Bot GPT 3.5]

The system of equations has one solution.

Answer 2

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

To determine how many solutions the system of equations has, we can solve the equations simultaneously and see the result.

The system of equations is:

y = -2x + 12 ...(Equation 1)
4y + x - 20 = 0 ...(Equation 2)

We can choose either the substitution method or the elimination method to solve the system of equations. Let's use the elimination method.

First, let's multiply both sides of Equation 1 by 4 to make the y-coefficient in both equations the same:

4y = -8x + 48 ...(Equation 1 multiplied by 4)
4y + x - 20 = 0 ...(Equation 2)

Now, let's subtract Equation 1 from Equation 2:

(4y + x - 20) - (-8x + 48) = 0
4y + x - 20 + 8x - 48 = 0
9x - 4y - 68 = 0 ...(Equation 3)

Now, let's multiply Equation 3 by -1:

-1(9x - 4y - 68) = 0
-9x + 4y + 68 = 0 ...(Equation 4)

Adding Equation 3 and Equation 4:

(9x - 4y - 68) + (-9x + 4y + 68) = 0
9x - 4y - 68 - 9x + 4y + 68 = 0
0 = 0

The result of adding Equation 3 and Equation 4 is 0 = 0, which means the two equations are equivalent and represent the same line. Therefore, the system of equations has infinitely many solutions.

The answer is: a) Infinitely many solutions.

Answer 3

[AI-generated answer — Explain Bot]

To determine the number of solutions for the system of equations, we can solve them simultaneously and observe the result.

The given system of equations is:

1) y = -2x + 12
2) 4y + x - 20 = 0

To start, we can rearrange the second equation to solve for x:

x = 20 - 4y

Now, we can substitute this value of x into the first equation:

y = -2(20 - 4y) + 12
y = -40 + 8y + 12
y - 8y = -40 + 12
-7y = -28
y = -28 / -7
y = 4

Next, substitute the y value back into the rearranged second equation to solve for x:

x = 20 - 4(4)
x = 20 - 16
x = 4

The solution to the system of equations is x = 4 and y = 4.

Since there is only one solution (x = 4, y = 4), the correct answer is: d) One solution.