three numbers: a, b, c
a+b+c = 11
2a +5b +6c = 32
3a -b = 22
<h3>Find</h3>a, b, c
<h3>Solution</h3>The equations can be represented by the augmented matrix
A graphing calculator gives the solution
(a, b, c) = (8, 2, 1)
The three numbers are 8, 2, and 1.
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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 <em>a</em>.
... a + 2 + 1 = 11
... a = 8 . . . . . . . . . . . as above
