SOLUTION: The sum of three numbers is 4. The first number minus the second plus 5 times the third is 20. The third plus 5 times the first plus the second is -8. Find the numbers.

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Question 597196: The sum of three numbers is 4. The first number minus the second plus 5 times the third is
20. The third plus 5 times the first plus the second is -8. Find the numbers.
Answer by math-vortex(648) About Me  (Show Source):
You can put this solution on YOUR website!
Hi, there--
To solve this problem, we define variables for each number. Then we create a system of three equations in three variables. Finally, we solve the system of equations to find the three numbers.
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[I] Define the variables.
Call the first number a.
Call the second number b.
Call the third number c.
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[II] Create three equations.
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The sum of the three numbers is 4. Using algebra, we write this relationship as
a%2Bb%2Bc=4
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The first number minus the second plus 5 times the third is
20. In algebra we write this relationship as
a-b%2B5c=20
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The third plus 5 times the first plus the second is -8. In algebra we write this relationship as
c%2B5a%2Bb=-8
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Rewrite the third equation to have the variables in alphabetical order.
5a%2Bb%2Bc=-8
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[III] Solve the system of equation.
I'll use the elimination and substitution methods here. You can also use a matrix to easily solve the system. Email me at my gmail address below if you want to see the matrix method, or repost this problem and ask for a matrix method solution.
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a%2Bb%2Bc=4
a-b%2B5c=20
5a%2Bb%2Bc=-8
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Add the 1st and 2nd equations together; solve to give an equation in a and c (NOTE: b+(-b)=0.)

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Subtract 6a%2B6c=12 from 2a%2B6c=24. Solve for a.
-4a=12
a=-3
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Substitute -3 for a in 2a%2B6c=24. Solve for c.
2%28-3%29%2B6c=24
%28-6%29%2B6c=24
6c=30
c=5
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Substitute -3 for a and 5 for c in the first equation. Solve for b.
a%2Bb%2Bc=4
%28-3%29%2Bb%2B%285%29=4
2%2Bb=4
b=2
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According to our algebra, the three numbers are -3, 2, and 5.
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[IV] Check your work.
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a%2Bb%2Bc=4
%28-3%29%2B2%2B5=4
4=4
Check!
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a-b%2B5c=20
%28-3%29-2%2B5%285%29=20
-5%2B25=20
20=20
Bingo!
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c%2B5a%2Bb=-8
%285%29%2B5%28-3%29%2B%282%29=-8
5-15%2B2=-8
-8=-8
Presto!
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That's it! Feel free to email me via gmail with any questions about the solution.
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Ms.Figgy
math.in.the.vortex@gmail.com