Question

How many solutions does this system have? y=5x-1 y=-3+7

Answer 1

I'll assume that's

$y=5x-1$

$y=-3x + 7$

Please let me know if that's wrong.

These are two lines in point slope form.   They'll meet at some x that makes the ys equal.


$5x-1 = -3x + 7$

$5x + 3x =1+ 7$

$8x= 8$

$x= 1$
 
$y=5x-1 = 5(1)-1 = 4$

$(x,y)=(1,4)$ is the only solution.

Exactly one solution.

We really only had to ensure the lines had two different slopes.  Then there's always exactly one solution.

If the lines had the same slope, they're either the same line (coincident) so an infinite number of solutions, all the points on that line, or they're parallel, no solutions.