Question

How many solutions does the system of equations have? y=−2x+12 4y+x−20=0

No solution No solution Two solutions Two solutions One solution One solution Infinitely many solutions Infinitely many solutions

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 system of equations, we need to solve the equations and analyze the result.

First, we'll solve the system of equations:

1. The first equation is given as y = -2x + 12.

2. Let's substitute the value of y from the first equation into the second equation:

4y + x - 20 = 0
4(-2x + 12) + x - 20 = 0
-8x + 48 + x - 20 = 0
-7x + 28 = 0
-7x = -28
x = -28 / -7
x = 4

3. Now substitute the value of x back into the first equation to find the value of y:

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

After solving the system, we found that x = 4 and y = 4 satisfy both equations, giving us a unique solution. Therefore, the system of equations has one solution.

Answer 3

[AI-generated answer — Explain Bot]

To determine the number of solutions for the given system of equations, we can use the method of substitution or elimination. In this case, we will use the method of elimination.

Given:
Equation 1: y = -2x + 12
Equation 2: 4y + x - 20 = 0

To eliminate one variable, let's multiply both sides of Equation 1 by 4:
4y = -8x + 48

Now, we have:
Equation 2: 4y + x - 20 = 0
Equation 3: 4y = -8x + 48

To eliminate y, we can subtract Equation 3 from Equation 2:
(4y + x - 20) - (4y) = (-8x + 48) - (-8x + 48)

Simplifying, we get:
x - 20 = 0

Solving for x, we get:
x = 20

Now, substituting this value of x back into Equation 1, we can find the corresponding value of y:
y = -2(20) + 12
y = -40 + 12
y = -28

Therefore, the solution for the system of equations is (x, y) = (20, -28). Since we have found a unique solution, the answer is: One solution.