My Math Forum [SOLVED] Algebra linear equation word problem

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February 2nd, 2012, 01:05 PM   #1
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[SOLVED] Algebra linear equation word problem

Quote:
 The difference between two numbers is 13. Twice the smaller plus three times the larger is 129. Find the two numbers.
I did:

x - y = 13

And now I don't know which one is the bigger or smaller one, so I have to choose from these 2 equations:

2x + 3y = 129
OR
2y + 3x = 129

Since in x - y = 13, it could be something like:
14 - 1 = 13, or -1 - -14, or something else. Either number can be smaller or larger.

Is there any way I can tell from the get-go which equation I have to use instead of trial-and-erroring both?

Anyway, here is path 1:

x - y = 13
2x + 3y = 129

Move over y in the 1st equation: x = y + 13
Plug in the second equation: 2(y + 13) + 3y = 129
2y + 26 + 3y = 129
5y = 103
y = 20.6

x = 20.6 + 13
x = 33.6

This is obviously wrong as the word problem said "twice the smaller" which means "2x" would have made 'x' the smaller number, but 'x' is larger than 'y' for this equation.

Path 2:

x - y = 13
2y + 3x = 129

x = y + 13

2y + 3(y + 13) = 129
2y + 3y + 39 = 129
5y = 90
y = 18

x = 18 + 13
x = 31

So now this makes sense, because 'y' was the smaller number in this case which is true.

Is there any way to avoid the unnecessary work in the future with more variables which would mean more guessing and checking?

February 2nd, 2012, 01:11 PM   #2
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Re: Algebra linear equation word problem

Quote:
Originally Posted by daigo
Quote:
 The difference between two numbers is 13. Twice the smaller plus three times the larger is 129. Find the two numbers.
I did: x - y = 13
And now I don't know which one is the bigger or smaller one, so I have to choose from these 2 equations:
But you DO know: x has to be greater than y, since difference is positive.

 February 2nd, 2012, 01:14 PM #3 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond Re: Algebra linear equation word problem Pick a variable as the smaller number and make your equations accordingly. Let x be the larger number and y be the smaller number. x - y = 13 (1) 3x + 2y = 129 (2) Multiply (1) by 2: 2x - 2y = 26, add this to (2): 5x = 155, x = 31, x - 13 = y, y = 18
 February 2nd, 2012, 04:10 PM #4 Global Moderator   Joined: Dec 2006 Posts: 21,019 Thanks: 2254 If the smaller is s, 2s + 3(s + 13) = 129, i.e. 5s + 39 = 129, so s = (129 - 39)/5 = 18 and s + 13 = 31.

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