Question

How many solutions does the system of equations have?

y=−2x+12

4y+x−20=0
(1 point)
Responses

One solution
One solution

Infinitely many solutions
Infinitely many solutions

Two solutions
Two solutions

No 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 the number of solutions for the given system of equations, we need to solve them simultaneously and see how many distinct solutions satisfy both equations.

First, let's solve the system of equations:

We have the following equations:

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

To solve this system, we can use the substitution method. We will solve Equation 1 for y and then substitute it into Equation 2.

From Equation 1, we have:
y = -2x + 12

Substituting this value of y into Equation 2, we get:
4(-2x + 12) + x - 20 = 0
-8x + 48 + x - 20 = 0
-7x + 28 = 0
-7x = -28
x = 4

Substituting the value of x = 4 into Equation 1, we get:
y = -2(4) + 12
y = 4

Therefore, the system of equations has one unique solution: x = 4, y = 4.

Hence, the answer is: One solution.

Answer 3

[AI-generated answer — Explain Bot]

To determine how many solutions a system of equations has, we can use the method of substitution or elimination to solve the system. Let's use the elimination method in this case:

First, let's write the given system of equations:

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

To use the elimination method, we need to multiply one or both equations by appropriate values in order to make the coefficient of one of the variables the same in both equations. In this case, we need to eliminate the variable x, so let's eliminate it by multiplying equation (1) by 4:

4y = -8x + 48 ----(3)

Now, we can combine equations (2) and (3) by adding them:

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

Simplifying the equation:

-7x + 4y + 28 = 0 ----(4)

Now, let's rearrange equation (4) to solve for y in terms of x:

4y = 7x - 28
y = (7x - 28)/4 ----(5)

Now we have the equations:

y = -2x + 12 ----(1)
y = (7x - 28)/4 ----(5)

Since these equations are in the form y = mx + c (where m represents the slope and c represents the y-intercept), we can graph them to determine the number of solutions.

Graphing these equations, we find that the lines are not parallel and intersect at a single point. Therefore, the system of equations has one solution.

Therefore, the correct answer is: One solution.