Gary is asked to find two whole numbers that have a sum of 9 and a product that is double the sum. He writes 7 and 2. Why is his answer incorrect?

Question

Good evening. Please forgive me as I am trying to assist my daughter with homework. I haven't done math in over 17 years. Thank you in advance for your assistance. 

Answer 1

Two whole numbers with a sum of 9 would be

x + y = 9

 

The product is double the sum so:

xy = 2(x+y) = 2(9) = 18

 

x + y = 9

xy = 18

 

NOTE:  I can see already that the numbers are 6 and 3.

7+2 = 9 but 7(2) = 14, not 18 so they don't fit the criteria given

 

To solve the 2 equations:

 

1)  x+y=9

2)  xy = 18

 

Solve for y in terms of x from equation 1.

y = 9-x

 

substitute 9-x in place of y in equation 2.

x(9-x) = 18

9x-x² = 18   add x² to both sides

9x = x² + 18   subtract 9x from both sides

0 = x² -9x + 18

 

We now have a quadratic that we can factor

 

x²-9x+18 = 0

Factors of 18 that add to -9 are (-6)(-3) so this

factors to:

 

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

x = 6 or x = 3

 

If x = 6 and x+y = 9 then y = 3

If x = 3 and x+y = 9 then y = 6

 

Either way the numbers are 3 and 6

 

 

 

 

 

Answer 2

Let the two numbers be x and y.

 

Given:

 

x + y = 9

xy = 2(x + y)

 

While his answer satisfies the top equation (7 + 2 = 9), it does not satisfy the bottom equation, since 7*2 ≠ 2(7 + 2).

 

But we can solve the two equations by substitution.

 

Substitute x + y = 9 from top equation into bottom equation and get

 

xy = 2*9 = 18

 

How many ways can you get xy = 18 with whole numbers such that x + y = 9 ?

 

xy = 1*18 = 18

xy =2*9 = 18

xy =3*6 = 18

 

Only the last one satisfies x + y = 9 and xy = 18 at the same time, so one number is 3, the other is 6.

 

Test the solution:

 

x + y = 3 + 6 = 9 (good)

xy = 3*6 = 18, and 2(x + y) = 2(3 + 6) = 2*9 = 18 (good)