SOLUTION: the difference of two positive numbers is 5 and the sum of their squares is 233. what are their numbers. i belive i am spoose to use substition but i'm not sure how, thanks

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Question 409186: the difference of two positive numbers is 5 and the sum of their squares is 233. what are their numbers. i belive i am spoose to use substition but i'm not sure how, thanks
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Let x = one of the positive numbers and y = the other positive number. Then "the difference of two positive numbers is 5" translates to:
x - y = 5
and "the sum of their squares is 233" translates to:
x%5E2+%2B+y%5E2+=+233

And yes, we will use the Substitution Method to solve this. This method starts with solving one equation for one of the variables. It does not matter which equation or which variable so let's find the easiest variable to solve for. Solving for x in the first equation looks easiest to me. It will be just one step: Adding y to each side:
x = y + 5
Next we go to the other equation and substitute in for the "solved for" variable:
%28y%2B5%29%5E2+%2B+y%5E2+=+233
Note the use of parentheses. It is an extremely good habit to use parentheses when substituting one expression for another, especially when the are multiple terms involved.

We now have a one variable equation to solve. First we simplify. Squaring y+5 can be done with FOIL or with the %28a%2Bb%29%5E2+=+a%5E2+%2B2ab%2Bb%5E2 pattern. I prefer to use the pattern:
%28y%29%5E2+%2B+2%28y%29%285%29+%2B+%285%29%5E2+%2B+y%5E2+=+233
which simplifies as follows:
y%5E2+%2B+10y+%2B+25+%2B+y%5E2+=+233
2y%5E2+%2B+10y+%2B+25+=+233
This is a quadratic equation so we want one side to be zero. Subtracting 233 we get:
2y%5E2+%2B+10y+-208+=+0
Now we factor (or use the Quadratic Formula). This factors fairly easily:
2%28y%5E2+%2B+5y+-104%29+=+0
2%28y-8%29%28y+%2B+13%29+=+0
From the Zero Product Property we know that this (or any) product can be zero only if one (or more) of the factors is zero. So:
2 = 0 or y-8 = 0 or y+13 = 0
There are no solutions to the first equation. But we can solve the other two:
y = 8 or y = -13.

Since y is supposed to be a positive number we will reject y = -13 as a solution. So y = 8. Now we find x. We can use the first equation:
x - y = 5
and substitute for y:
x - (8) = 5
Adding 8 to each side we get:
x = 13.

So the two positive numbers are 8 and 13.