How many solutions does the system have?
y=x+5
y=-5x+1

There is only ONE solution; which is:  (-⅔, 4 ⅓) .

Explanation:

Given:

y = x + 5 ; 
y = -5x +1 ; 

How many solutions does the system have?

Note: -5x + 1 = 1 − 5x

So, x + 5 = -5x + 1 ; 

Add "5x" to  EACH SIDE; and subtract "5" from EACH SIDE;

     x + 5 + 5x − 5 = -5x + 1 + 5x − 5 ;

to get:  6x = -4  ;

Divide each side of the equation by "6" ; to isolate "x" on one side of the equation ; and to solve for "x" ; 
  
     6x / 6 = -4/6 ;
 
         x =  -⅔ ;

There is only one solution.

Let us double check by plugging this value for "x" into EACH equation; to see if the same value for "y" holds true:

First equation:

y = x + 5;

y = (-2/3) + 5 = 5 − ⅔ = 15/3 − 2/3 = (15 − 2) / 3 = 13/3;


So; in the second equation, does "y = 13.3 ; when "x = -⅔ " ?

y = -5x + 1 ;


 y = -5*(-⅔)  + 1 = (-5/1)*(-2/3)  + 1   = (-5*-2) /(1*3)  + 1 ;
 
                              =  10/3 + 1 = 10/3 + 3/3  = (10 + 3) / 3  = 13/3 ;

Yes!  So; there is only ONE solution. 

x = -⅔ ; y = 13/3 ;   13/3 can be written as: "4 ⅓",

So; the answer is:  There is only ONE solution; which is:
 
(-⅔, 4 ⅓) .