Question

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

Answer 1

using y=x+5 to substitute for y in y=-5x-1,

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

i think its one
because if you make them equal to each other (cause theyre both y's), you only get one answer which is -1

Answer 2

Answer: There is only ONE solution; which is: (-⅔, 4 ⅓) .
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Step-by-step explanation:
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Given:

y = x + 5 ;
y = -5x +1 ;

How many solutions does the system have?
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Note: -5x + 1 = 1 − 5x

So, x + 5 = -5x + 1 ;

Add "5x" to EACH SIDE; and subtract "5" from EACH SIDE;
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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:
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First equation:

y = x + 5;

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

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

y = -5x + 1 ;
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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 ;
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Yes! So; there is only ONE solution.

x = -⅔ ; y = 13/3 ; 13/3 can be written as: "4 ⅓",
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So; the answer is: There is only ONE solution; which is:

(-⅔, 4 ⅓) .
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