The sum of two numbers is 38. The greater number is 8 more than the other number. Find each number. Use a system of linear equations to justify your answer

The smaller number is

3

(The other number is therefore

5

)

Explanation:

We could use

x

and

y

to represent the two numbers, but it is possible to form an equation using only one variable.

If two numbers add up to 8, and we let one of them be

x

, the other one will be the difference, which is:

8

-

x

The sum of their squares is

34

x

2

+

(

8

-

x

)

2

=

34

←

(

a

-

b

)

2

=

a

2

-

2

a

b

+

b

2

x

2

+

64

-

16

x

+

x

2

=

34

2

x

2

-

16

x

+

64

-

34

=

0

2

x

2

-

16

x

+

30

=

0

←

:

2

x

2

-

8

x

+

15

=

0

←

find factors of 15 which add to 8.

(

x

-

3

)

(

x

-

5

)

=

0

Setting factor equal to

0

gives:

x

=

3

and

x

=

5

Check:

3

+

5

=

8

3

2

+

5

2

=

9

+

25

=

34