Question

How many solutions does the system of equations have? y=−2x+12 4y+x−20=0 (1 point) Responses No solution No solution Infinitely many solutions Infinitely many solutions One solution One solution Two solutions

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the number of solutions, we can solve the system of equations.

By substitution, we have:

-y = 2x - 12 (from the first equation)

x = 4y - 20 (from the second equation)

Substituting the value of x from the second equation into the first equation, we get:

-(4y - 20) = 2x - 12

-4y + 20 = 2x - 12

-4y = 2x - 32

Dividing the equation by 2:

-2y = x - 16

Now we have two equations:

-y = 2x - 12

-2y = x - 16

Rearranging the second equation:

x - 2y = 16

We can now see that the two equations are equivalent; they represent the same line. Therefore, the system of equations has infinitely many solutions.

So, the answer is:

Infinitely many solutions

Answer 2

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

To determine the number of solutions the system of equations has, we need to solve it. Let's start by rearranging the second equation:

4y + x - 20 = 0

Rearranging this equation, we get:

x = 20 - 4y

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

y = -2x + 12

y = -2(20 - 4y) + 12

Simplifying, we get:

y = -40 + 8y + 12

Combining like terms, we have:

7y = -28

Dividing both sides by 7, we find:

y = -4

Now we can substitute this value of y back into the second equation to find x:

x = 20 - 4(-4)

x = 20 + 16

x = 36

So we have found one solution for the system of equations, which is (x, y) = (36, -4).

Therefore, the system of equations has one solution.

Answer 3

[AI-generated answer — Explain Bot]

To determine the number of solutions for a system of equations, we can solve the equations to find the values of x and y and see how they relate.

The system of equations is:

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

First, let's solve equation 1 for y:

y = -2x + 12

Now, substitute this expression for y in equation 2:

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

Simplify the equation:

-8x + 48 + x - 20 = 0

Combine like terms:

-7x + 28 = 0

Subtract 28 from both sides:

-7x = -28

Divide both sides by -7:

x = 4

Now, substitute this value of x back into equation 1 to solve for y:

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

So, the values of x and y that satisfy both equations are x = 4 and y = 4.

Therefore, the system of equations has one solution.

Answer 4

are you sure

Answer 5

[AI-generated answer — Bot GPT 3.5]

Apologies for the confusion in my previous responses. Let's re-evaluate the system of equations to determine the number of solutions.

The system of equations is:

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

To solve this system, we can substitute the value of y from equation 1 into equation 2:

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

Simplify the equation:

-8x + 48 + x - 20 = 0

Combine like terms:

-7x + 28 = 0

Subtract 28 from both sides:

-7x = -28

Divide both sides by -7:

x = 4

Now, substitute this value of x back into equation 1 to solve for y:

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

Thus, we have found that the values of x = 4 and y = 4 satisfy both equations in the system.

Therefore, the system of equations has one solution.

Answer 6

correct

Answer 7

[AI-generated answer — Bot GPT 3.5]

Great! I apologize for the confusion earlier. The correct answer is that the system of equations has one solution. Thank you for pointing that out. If you have any more questions, feel free to ask.