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Question 560696: The sum of three integers is 53. If the first integer is doubled, the sum is 67. If the second integer is doubled, the sum is 69. Find the integers.
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Let the three unknown integers be represented by a, b, and c.
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The sum of the three equals 53. In equation form this is:
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a + b + c = 53 <---- this is equation #1
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If you double the first integer (which is a) and add the other two integers, the result is 67. In equation form this is:
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2a + b + c = 67 <---- this is equation #2
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Finally, if you double the second integer (which is b) and add the other two integers, the result is 69. In equation form this is:
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a + 2b + c = 69 <---- this is equation #3
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Equations #1 and #3 have both a and c in common. Let's subtract equation #1 from equation #3.
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a + 2b + c = 69 <----equation #3
a + b + c = 53 <----equation #1 (to be subtracted from eqn #3 vertically)
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When you subtract in vertical columns notice that the a and c terms will drop out. After the subtraction you will be left with:
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b = 16
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and now we have one of the unknown constants.
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Now let's double equation #1 by multiplying all of its terms on both sides by 2. When you do that the equation becomes:
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2a + 2b + 2c = 106 <---- equation #1 doubled
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From this subtract equation #2 as shown below:
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2a + 2b + 2c = 106 <---- equation #1 doubled
2a + b + c = 67 <---equation #2 to be subtracted.
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After subtracting vertically in columns the 2a minus 2a disappear and you are left with b + c = 39.
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But we know that b = 16. So in the reduced equation b + c = 39, when we substitute 16 for b we have:
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16 + c = 39
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Subtract 16 from both sides and you are left with:
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c = 23
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Now we can go to equation #1 and substitute 16 for b and 23 for c and the equation becomes:
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a + 16 + 23 = 53
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Add the two constants on the left side and the equation becomes:
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a + 39 = 53
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Subtract 39 from both sides of this equation and the result is:
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a = 14
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So in summary, we now know that a = 14, b = 16, and c = 23.
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You can substitute these values for a, b, and c in the original 3 equations and you should find that the left side of each equation will equal the right side, thereby validating the answers.
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I hope this helps you understand the way you can do this problem. The way we wrote the original 3 equations is where you start. How you choose to eliminate variables between pairs of these equations is up to you because there are other combinations that you can use. The above process is just one of the ways it could be done. At least now that you know the answers, you can try some of the other ways that pairs of equations can be subtracted to eliminate variables. These other ways should still result in the same answers. Good luck ...
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