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Question 1162557: The sum of three numbers is 17. The sum of twice the first number, 3 times the second number, and 4 times the third number is 49. The difference between 3 times the first number and the second number is 21. Find the three numbers.
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52914)   (Show Source):
You can put this solution on YOUR website! .
x + y + z = 17 (1)
2x + 3y + 4z = 49 (2)
3x - y = 21 (3)
It can be solved by the Substitution method, or by the Elimination method, or by using determinants (the Cramer's rule).
To simplify the life, I will use one of the free of charge online solvers
https://matrix.reshish.com/cramSolution.php
(Cramer's rule).
The answer is x = 8, y = 3, z = 6.
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If you wish to learn about the methods of solution, look into the lessons
- Solving systems of linear equations in 3 unknowns by the Substitution method,
- BRIEFLY on solving systems of linear equations in 3 unknowns by the Substitution method,
- Solving systems of linear equations in 3 unknowns by the Elimination method and
- BRIEFLY on solving systems of linear equations in 3 unknowns by the Elimination method
- Determinant of a 3x3 matrix
- Co-factoring the determinant of a 3x3 matrix
- HOW TO solve system of linear equations in three unknowns using determinant (Cramer's rule)
- Solving systems of linear equations in three unknowns using determinant (Cramer's rule)
- Solving word problems by reducing to systems of linear equations in three unknowns
in this site.
Also, you have this free of charge online textbook in ALGEBRA-II in this site
- ALGEBRA-II - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this online textbook under the topic
"3x3-Matrices, determinants, Cramer's rule for systems in three unknowns"
In addition, there are many free of charge SOLVERS on a Cramer's rule in the Internet.
One of such popular solvers is in this site under the link
https://www.algebra.com/algebra/homework/Matrices-and-determiminant/cramers-rule-3x3.solver
https://www.algebra.com/algebra/homework/Matrices-and-determiminant/cramers-rule-3x3.solver
Answer by greenestamps(13215)  (Show Source):
You can put this solution on YOUR website!
This problem can give you good practice for setting up a problem for solving using formal algebra, and for then solving the problem. You should learn how to do that.
(1) The sum of three numbers is 17.
(2) The sum of twice the first number, 3 times the second number, and 4 times the third number is 49.
(3) The difference between 3 times the first number and the second number is 21.
There are numerous ways of solving that system of three equations to find the values of a, b, and c. Because of that, I will only show one possible way to start on that task. You can finish the way I show; or you can use a completely different path if you want.
With the form of the three equations, my preference would be to solve (3) for b and substitute the result in (1) and (2); that will give me two equations in a and c.
Substituting in (1):
(4) 
Substituting in (2):
(5) 
Solve (4) and (5)....
This problem also gives you an excellent opportunity to get some good mental exercise by finding a solution using logical reasoning and some basic arithmetic. Here is how it might go.
To start, the nature of the question suggests that the three numbers are positive integers. So let's assume that.
The sum of the three numbers is 17; and the difference between 3 times the first number and the second number is 21. With our assumption that the three numbers are positive integers, the smallest possible value for the first number is 8.
If the first number is 8, the second number is 3 (to make the difference between 3 times the first number and the second number 21). And, since the sum of the three number is 17, that means the third number is 6.
The possible solution (8,3,6) satisfies (1) and (3); and checking it in (2) we see that requirement is also satisfied.
So we have found the solution by logical reasoning, without formal algebra.
ANSWER: The three numbers are 8, 3, and 6.
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