How many solutions does the system of equations have?

Question

y=−2x+12
4y+x−20=0

Answer 1

To solve this system of equations, we can use substitution or elimination method.

1) Substitution method:
First, in the first equation, we can solve for y in terms of x:
y = -2x + 12

Now, substitute this value of y in the second equation:
4(-2x + 12) + x - 20 = 0
-8x + 48 + x - 20 = 0
-7x + 28 = 0
-7x = -28
x = 4

Now, substitute this value of x back into the first equation:
y = -2(4) + 12
y = 8

So, the solution to this system of equations is x = 4 and y = 8. Therefore, the system has exactly one solution.

2) Elimination method:
Multiply the first equation by 4:
4y = -8x + 48

Now, rewrite the second equation:
4y + x - 20 = 0

Subtract the first equation from the second equation:
(4y + x - 20) - (-8x + 48) = 0
4y + x - 20 + 8x - 48 = 0
4y + 9x - 68 = 0

Now, solve for y in terms of x:
4y = -9x + 68
y = (-9/4)x + 17

From these equations, we can see that the equations of the lines have different slopes, so they will intersect at a single point. Therefore, the system has exactly one solution.