Determine how many solutions exist for the system of equations.
y=2x-9
y=-x+6

2x-9=-x+6
+x      +x
3x-9=6
    +9 +9
3x=15

x=5

therefore the equations have one solution (x=5)

Since we want to find solutions that will work for both of the equations, we must  first assume that x and y are both the same in both so

y=2x-9
y=-x+6
we make the second equation negative by multiplying the whole thing by -1 and get
-y=x-6

add this to the first equation
y-y=x+2x-9-6
0=3x-15
add 15 to both sides
15=3x
divide both sides by 3
5=x

subsitute 5=x for x in the second equation

y=-x+6
y=-5+6
y=1