Question 1163244

Hi

The sum of the ages of Ruth Selma and Tom is one year more than 3 times toms age.the sum of Tom and Selma ages is 2 years more than 3 times Ruth's age.
The sum of Ruth and toms ages is the same as the sum of selmas age and half of ruths age.  How old is each one.

Thanks
<pre>Let Ruth's, Selma's, and Tom's ages, be R, S, and T, respectively
Then we get: R + S + T = 3T + 1 ===> R + S - 2T = 1 ------ eq (i)
             S + T = 3R + 2 ======> 3R - S - T = - 2 ----- eq (ii)
             {{{matrix(2,3, R + T, "=", S + R/2, 2R + 2T, "=", 2S + R)}}}
             R - 2S + 2T = 0 ------ eq (iii)
Multiply eq (ii) by 2 to get eq (iv) in the following system: {{{matrix(3,6, R + S - 2T, ""="", 1, "------", eq, "(i)", R - 2S + 2T, ""="", 0, "------", eq, "(iii)", 6R - 2S - 2T, ""="", - 4, "------", eq, "(iv)")}}}
2R - S = 1 ----- Adding eqs (i) + (iii) ----- eq (v)
7R - 4S = - 4 -- Adding eqs (iii) + (iv) ---- eq (vi)
8R - 4S = 4 ---- Multiplying eq (v) by 4 --- eq (vii)
Ruth, or {{{highlight_green(matrix(1,4, R, "=", 8, years-old))}}} ----- Subtracting eq (vi) from eq (vii)

2(8) - S = 1 ----- Substituting 8 for R in eq (v)
16 - S = 1
- S = 1 - 16
- S = - 15
Selma, or {{{highlight_green(matrix(1,6, S, "=", (- 15)/(- 1), "=", 15, years-old))}}} 

3R - S - T = - 2 ------ eq (ii)
3(8) - 15 - T = - 2 --- Substituting 8 for R, and 15 for A in eq (ii)
24 - 15 - T = - 2
- T = - 2 - 9
- T = - 11
Tom, or {{{highlight_green(matrix(1,6, T, "=", (- 11)/(- 1), "=", 11, years-old))}}}</pre>