Question 333707
In a group of five friends, the sums of the ages of each group of four of them are 124, 128, 130, 136, and 142. What is the age of the youngest?
<pre><b>
Let the five ages be A, B, C, D, and E.

A + B + C + D = 124
A + B + C + E = 128
A + B + D + E = 130
A + C + D + E = 136
B + C + D + E = 142

We consider that as this system, putting in all the 
understood 1's and 0's

1A + 1B + 1C + 1D + 0E = 124
1A + 1B + 1C + 0D + 1E = 128
1A + 1B + 0C + 1D + 1E = 130
1A + 0B + 1C + 1D + 1E = 136
0A + 1B + 1C + 1D + 1E = 142

Now we make the augmented matrix:

{{{(matrix(5,7,
1,  1,  1,  1,  0, "|", 124,
1,  1,  1,  0,  1, "|", 128,
1,  1,  0,  1,  1, "|", 130,
1,  0,  1,  1,  1, "|", 136,
0,  1,  1,  1,  1, "|", 142))}}}

Using my TI-84 I get for the rref of that matrix:

{{{(matrix(5,7,
1,  0,  0,  0,  0, "|", 23,
0,  1,  0,  0,  0, "|", 29,
0,  0,  1,  0,  0, "|", 35,
0,  0,  0,  1,  0, "|", 37,
0,  0,  0,  0,  1, "|", 41))}}}

A=23, B=29, C=35, D=37, E=41.

The youngest age is A, which is 23. 

Edwin</pre>