Question 980579


Let &nbsp;<B>a</B>&nbsp; be the number of years Adrian has been teaching.

Let &nbsp;<B>b</B>&nbsp; be the number of years Betty has been teaching.

Let &nbsp;<B>c</B>&nbsp; be the number of years Charlie has been teaching.


Then we have the system of three equations in three unknowns


a + b &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;= 36, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;b + c = 22, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2)

a + &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;c = 28. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(3)


It is very special kind of system, &nbsp;and we will apply special method to solve it.

Add all three equations &nbsp;(left sides and right sides separately).

You will get 


2(a + b + c) = 86, &nbsp;&nbsp;&nbsp;&nbsp;or


a + b + c = 43. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(4)


Now, &nbsp;distract the equation &nbsp;(1)&nbsp; from the equation &nbsp;(4). &nbsp;You will get 


c = 43 - 36 = 7.


Hence, &nbsp;Charlie has been teaching for &nbsp;7&nbsp; years.

Next, &nbsp;distract the equation &nbsp;(2)&nbsp; from the equation &nbsp;(4). &nbsp;You will get 


a = 43 - 22 = 21.


Hence, &nbsp;Adrian has been teaching for &nbsp;21&nbsp; years.


Next, &nbsp;distract the equation &nbsp;(3)&nbsp; from the equation &nbsp;(4). &nbsp;You will get 


b = 43 - 28 = 15.


Hence, &nbsp;Betty has been teaching for &nbsp;15&nbsp; years.


For more problems similar to this one see the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/The-trick-to-solve-some-word-problems-with-three-and-more-unknowns.lesson>The tricks to solve some word problems with three and more unknowns using mental math</A>&nbsp; in this site.