Question

The sum of three numbers is 11 The sum of twice the first number five times the second number and six times the third number is 32 The difference between three times the first number in the second number is 22 find the three numbers

Answer 1

Given

three numbers: a, b, c

a+b+c = 11

2a +5b +6c = 32

3a -b = 22

Find

a, b, c

Solution

The equations can be represented by the augmented matrix

$\left[\begin{array}{ccc|c}1&1&1&11\\2&5&6&32\\3&-1&0&22\end{array}\right]$

A graphing calculator gives the solution

(a, b, c) = (8, 2, 1)

The three numbers are 8, 2, and 1.

_____

If you want to solve this by hand, you could use Cramer's rule, or you could do the row operations by and. For example, subtract twice the first equation from the second to get

... 3b +4c = 10

Subtract 3 times the first equation from the third to get

... -4b -3c = -11

These two equations can be solved by your favorite method to find

... b = (-44 +30)/(-16 +9) = -14/-7 = 2 . . . . . using Cramer's rule

... c = (-40 +33)/-7 = 1

Then the first equation can be used to find a.

... a + 2 + 1 = 11

... a = 8 . . . . . . . . . . . as above