SOLUTION: Part of $20,000 is invested at 5% and the rest at 3%. How much is invested at each rate if the total income is 4.5% of the total investment?

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Question 111380: Part of $20,000 is invested at 5% and the rest at 3%. How much is invested at each rate if the total income is 4.5% of the total investment?
Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!
Let x=amount invested for 5%, y=amount for 3%

So we know the sum of the parts is the total $20,000. So this means

x%2By=20000


Also we know that 5% of x amount of dollars plus 3% of y amount of dollars is 4.5% of $20,000 (which is $900)

So

0.05x%2B0.03y=900


5x%2B3y=90000 Multiply both sides by 100 to make every number a whole number


Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=20000
5%2Ax%2B3%2Ay=90000

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and 5 to some equal number, we could try to get them to the LCM.

Since the LCM of 1 and 5 is 5, we need to multiply both sides of the top equation by 5 and multiply both sides of the bottom equation by -1 like this:

5%2A%281%2Ax%2B1%2Ay%29=%2820000%29%2A5 Multiply the top equation (both sides) by 5
-1%2A%285%2Ax%2B3%2Ay%29=%2890000%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
5%2Ax%2B5%2Ay=100000
-5%2Ax-3%2Ay=-90000

Notice how 5 and -5 add to zero (ie 5%2B-5=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%285%2Ax-5%2Ax%29%2B%285%2Ay-3%2Ay%29=100000-90000

%285-5%29%2Ax%2B%285-3%29y=100000-90000

cross%285%2B-5%29%2Ax%2B%285-3%29%2Ay=100000-90000 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

2%2Ay=10000

y=10000%2F2 Divide both sides by 2 to solve for y



y=5000 Reduce


Now plug this answer into the top equation 1%2Ax%2B1%2Ay=20000 to solve for x

1%2Ax%2B1%285000%29=20000 Plug in y=5000


1%2Ax%2B5000=20000 Multiply



1%2Ax=20000-5000 Subtract 5000 from both sides

1%2Ax=15000 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ax=%2815000%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.


x=15000 Multiply the terms on the right side


So our answer is

x=15000, y=5000

which also looks like

(15000, 5000)

Notice if we graph the equations (if you need help with graphing, check out this solver)

1%2Ax%2B1%2Ay=20000
5%2Ax%2B3%2Ay=90000

we get



%0D%0A++drawing%28+500%2C+600%2C+-15010%2C+15010%2C+-5010%2C+5010%2C%0D%0A++++graph%28+500%2C+600%2C+-15010%2C+15010%2C+-5010%2C+5010%2C+%2820000-1%2Ax%29%2F1%2C+%2890000-5%2Ax%29%2F3+%29%2C%0D%0A++++blue%28++circle%28+15000%2C+5000%2C+200.133333333333+%29+%29+%0D%0A++%29%0D%0A++ graph of 1%2Ax%2B1%2Ay=20000 (red) 5%2Ax%2B3%2Ay=90000 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (15000,5000). This verifies our answer.


note: ignore the graph. It didn't turn out right

So $15,000 was invested at 5% and $5,000 was invested at 3%