To find how many solution our system has, we are going to find the slope of each equation.
[tex]y=x-2[/tex] equation (1)
[tex]-x+y=-5[/tex] equation (2)
To find the slope, we are going to express both equations in the form: [tex]y=mx+b[/tex]
where
[tex]m[/tex] is the slope
Notice that equation (1) is already in the form [tex]y=mx+b[/tex]; from equation (1) we can infer that [tex]m=1[/tex]
To express equation (2) in the form [tex]y=mx+b[/tex], we are going to add [tex]x[/tex] to both sides of the equation:
[tex]-x+x+y=x-5[/tex]
[tex]y=x+5[/tex]
Now, we can infer that the slope of equation (2) is [tex]m=1[/tex]. Since both equations have the same slope, we are dealing with parallel lines; parallel lines don't intercept, so the system has no solutions.
We can conclude that the correct answer is the first choice: no solutions
