Question

How many solutions does the system of equations have?

y = −2x +12
4y + x −20 = 0

Answer 1

all you need to know is that the slopes are different.
Therefore the lines will intersect in a point -- the solution.

Answer 2

To solve this system of equations, we can use substitution. We know that y in the second equation is equal to -2x + 12 based on the first equation, so we can substitute that into the second equation:

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

Simplifying this equation, we get:

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

Now that we know x is 4, we can substitute that back into either equation to solve for y. Let's use the first equation:

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

So the solution to this system of equations is (4, 4). Therefore, there is only one solution.

Answer 3

Yes, that is correct. If the slopes of two lines are different, then the lines intersect at exactly one point, giving us a unique solution for the system of equations.