Question

How many solutions does the system of equations have?

y=−2x+12
4y+x−20=0

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the number of solutions for the system of equations, we can solve them to determine if they intersect at a single point (1 solution), parallel lines (no solution), or coincident lines (infinitely many solutions).

We'll start by rearranging the second equation to isolate x:
4y + x - 20 = 0
x = 20 - 4y

Now we substitute this expression for x into the first equation:
y = -2x + 12
y = -2(20 - 4y) + 12
y = -40 + 8y + 12
-7y = -28
y = 4

Finally, we substitute this value of y into the expression we found for x:
x = 20 - 4y
x = 20 - 4(4)
x = 4

Therefore, the system of equations has a unique solution, which is (4, 4).

Answer 2

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

To determine the number of solutions, let's solve the system of equations.

1. Let's start by rearranging the second equation to isolate one variable.
4y + x - 20 = 0 --> x = 20 - 4y

2. Now substitute the expression for x in the first equation.
y = -2x + 12 --> y = -2(20 - 4y) + 12

3. Simplify the equation by distributing and combining like terms.
y = -40 + 8y + 12 --> y - 8y = -40 + 12
-7y = -28 --> y = -28 / -7
y = 4

4. Substitute the value of y back into the equation for x.
x = 20 - 4y --> x = 20 - 4(4) --> x = 20 - 16 --> x = 4

5. After solving the equations, we find that the system has exactly one solution, which is (4, 4).

Answer 3

[AI-generated answer — Explain Bot]

To determine the number of solutions the system of equations has, we need to solve them simultaneously.

First, let's rewrite the second equation to make it easier to work with:

4y + x - 20 = 0

Next, we can use the substitution method by substituting y = -2x + 12 from the first equation into the second equation:

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

Simplifying the equation:

-8x + 48 + x - 20 = 0

Combining like terms:

-7x + 28 = 0

Now, 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 either of the original equations to find y. Let's use the first equation:

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

So, the solution of the system of equations is x = 4 and y = 4.

Since we obtained a unique solution for the system of equations, it has only one solution.