SOLUTION: 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 t

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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
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39630) About Me  (Show Source):
You can put this solution on YOUR website!
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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.
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Ruth        r
Selma       x
Tom         t

system%28r%2Bx%2Bt=1%2B3t%2Cx%2Bt=2%2B3r%2Cr%2Bt=x%2Br%2F2%29
-
system%28r%2Bx-2t=1%2C-3r%2Bx%2Bt=2%2C2r%2B2t=2x%2Br%29

system%28r%2Bx-2t=1%2C3r-x-t=-2%2Cr-2x%2B2t=0%29

From here or earlier, methods can vary, but choosing E1+E3, this gives 2r-x=1. This means, x=2r-1.

Substituting for the expression for x,
system%28r%2B2r-1-2t=1%2C3r-2r%2B1-t=-2%2Cr-2%282r-1%29%2B2t=0%29

system%283r-2t=2%2Cr-t=-3%2Cr-4r%2B2%2B2t=0%29

system%283r-2t=2%2Cr-t=-3%2C-3r%2B2t=-2%29------------here, E1 and E3 are equivalent, so,...

system%283r-2t=2%2Cr-t=-3%29

.

Answer by MathTherapy(10557) About Me  (Show Source):
You can put this solution on YOUR website!

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
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%282%2C3%2C+R+%2B+T%2C+%22=%22%2C+S+%2B+R%2F2%2C+2R+%2B+2T%2C+%22=%22%2C+2S+%2B+R%29
             R - 2S + 2T = 0 ------ eq (iii)
Multiply eq (ii) by 2 to get eq (iv) in the following system: 
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%28matrix%281%2C4%2C+R%2C+%22=%22%2C+8%2C+years-old%29%29 ----- 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  

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