SOLUTION: Solve the following problem using Elimination method. 1. how many gallons of 20% alcohol solution and 50% alcohol solution must be mixed to get 9 gallons of 30% alcohol solution

Algebra ->  Expressions-with-variables -> SOLUTION: Solve the following problem using Elimination method. 1. how many gallons of 20% alcohol solution and 50% alcohol solution must be mixed to get 9 gallons of 30% alcohol solution      Log On


   

Question 825251: Solve the following problem using Elimination method.
1. how many gallons of 20% alcohol solution and 50% alcohol solution must be mixed to get 9 gallons of 30% alcohol solution?
2. an airplane flying in the same direction as the wind can cover a certain distance in 2hrs. the return trip is against the direction of wind and takes 2.5 hrs. what are the speeds of the wind and the plane, if the one-way distance is 960km?
3. it takes a boat 2hrs to travel 24km downstream and 3hrs to travel 18km upstream. what are the speeds of the boat in still water and of the current of the river?
Answer by TimothyLamb(4379) About Me  (Show Source):
You can put this solution on YOUR website!
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This post contains three different questions.
I will answer the first question.
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question:
How many gallons of 20% alcohol solution and 50% alcohol solution must be mixed to get 9 gallons of 30% alcohol solution?
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x = gallons of 20% solution
y = gallons of 50% solution
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20x + 50y = 30*9
x + y = 9
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put the system of linear equations into standard form
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20x + 50y = 270
x + y = 9
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copy and paste the above linear system in standard form into this matrix-method solver:
https://sooeet.com/math/system-of-linear-equations-solver.php
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solution by matrix method:
x = gallons of 20% solution = 6
y = gallons of 50% solution = 3
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system matrix
20 50
1 1
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inverse of system matrix
-0.033333333 1.6666667
0.033333333 -0.66666667
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determinant of system matrix = -30
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now the solution by elimination:
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20x + 50y = 270
(x + y = 9)*(-20)
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20x + 50y = 270
-20x - 20y = -180
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add the system:
30y = 90
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y = 3
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x + y = 9
x + 3 = 9
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x = 6
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solution by elimination method:
x = gallons of 20% solution = 6
y = gallons of 50% solution = 3
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NOTE:
solution by matrix method agrees with solution by elimination method
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