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

In order to determine the number of solutions for this system of equations, we need to find the intersection point of the two lines represented by the equations.

First, let's solve the system of equations:

Substitute the value of y from the first equation into the second equation:
4*(-2x + 12) + x - 20 = 0
-8x + 48 + x - 20 = 0
-7x + 28 = 0
-7x = -28
x = 4

Substitute the value of x into the first equation to find y:
y = -2*(4) + 12
y = -8 + 12
y = 4

Therefore, the system of equations has one solution.

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

The given system of equations is:

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

To solve the system, we can use the method of substitution or elimination. Let's use the substitution method.

First, let's solve equation 1) for y:

y = -2x + 12

Next, substitute this expression for y in equation 2):

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

Now, simplify the equation:

-8x + 48 + x - 20 = 0
-7x + 28 = 0

Next, isolate x by moving the constant term to the other side:

-7x = -28

Divide both sides by -7 to solve for x:

x = 4

Now that we have the value of x, substitute it back into equation 1) to solve for y:

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

Therefore, the system of equations has a unique solution, which is x = 4 and y = 4.

In summary, the system of equations has a single solution.

To determine the number of solutions for the system of equations, we need to solve them simultaneously. Let's begin:

For the first equation: y = -2x + 12...

And for the second equation: 4y + x - 20 = 0...

To solve this system, we will use the substitution method, which involves replacing one variable with an expression involving the other variable. We can substitute y in the second equation with the expression -2x + 12 since y = -2x + 12:

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

Now we can simplify the equation:

-8x + 48 + x - 20 = 0,
-7x + 28 = 0,
-7x = -28,
x = 4.

Now that we found the value of x, we can substitute it back into one of the original equations to solve for y:

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

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

Therefore, the system of equations has one solution.

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