pls help with this question

Question

The sum of two integers is –5. When 10 times the larger number is subtracted from the square of the smaller number, the result is 34. Find the system of equations that could be used to determine the two integers. Solve for the integers.

Answer 1

The problem gives us these equations:

a + b = -5;

b^2 - 10 a = 34 (we choose a to be bigger)

The first trick we want to do is to use our first equation to solve for a:

a = -5 -b;

Next, let's plug it in to our second equation:

b^2 - 10(-5-b) = 34;

b^2 +10b + 50 = 34;

Let's complete the square so:

b^2 +10b + 25 + 25 = 34;

(b+5)^2 + 25 = 34;

(b+5)^2 = 9;

(b+5) = +3 or -3

so b = -2 or -8

If b is -2 then a must be -3 (plug into first equation: a + b = -5).

Similarly, if b is -8, a must be 3.

a = -3, b = -2 (ignore this solution since a must be bigger than b)

a = 3, b = -8

Answer 2

L+S = -5, L= -5-S

S^2 - 10L = 34

S^2 -10(-5-S) = 34

S^2 +50 +10S = 34

S^2 +10S + 16 = 0

(S+8)(S+2) = 0

S= -2 or -8

L=-5-S = -5+2 or -5+8 = -3 or +3

either (S,L) = (-2,-3) or (-8,+3) but ignore the 1st solution as L<S

larger number is +3 with smaller number = -8