Two positive numbers have a sum of 8 and their product is equal to the larger number plus 10

There are 2 sets of numbers that work...

x = 3, y = 5

and

x = 6, y = 2

Step-by-step explanation:

Let x be one number, and let y be the other number.  We have 2 equations...

x + y = 8    (two positive numbers have a sum of 8)

xy = y + 10   (their product is equal to the larger number plus 10)

solve the first equation for y, and substitute that into the second equation...

y = 8 - x

  x(8 - x) = 8 - x + 10

now solve for x...

  8x - x² = 18 - x  

This is a quadratic, so get everything to one side so it's equal to zero...

   -x² + 9x - 18 = 0   (add x and subtract 18 from both sides)

Now solve for x...

     x² - 9x + 18 = 0    (divide both sides by -1)

         (x - 6)(x - 3) = 0    (factor)

so

x - 6 =  0  becomes   x = 6    (add 6 to both sides)

and

x - 3 = 0  becomes x = 3      (add 3 to both sides

If x is 3, then y = 5     (3 + 5 is 8, and 3(5) = 5 + 10, both equations hold up)

If x is 6, then y = 2    (6 + 2 is 8, and 6(2) = 2 + 10, both equation hold up)

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