Question 1201076: Formulate a system of equations for the situation below and solve.
Michael Perez deposited a total of $2000 with two savings institutions. Bank A pays interest at the rate of 5%/year, whereas Bank B pays interest at the rate of 7%/year. If Michael earned a total of $108 in interest during a single year, how much did he deposit in each institution?
Bank A $
Bank B $
Found 3 solutions by josgarithmetic, greenestamps, ikleyn:
Answer by josgarithmetic(39615)   (Show Source):
You can put this solution on YOUR website! Bank A, 5% rate
Bank B, 7% rate
Initial total deposit $2000
one-year of interest $108
If he deposited g dollars at 7% rate, then 2000-g was at the 5% rate.
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Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
The response from the other tutor shows a setup for solving the problem using a single equation; you asked for a solution using a system of equations.
Let x = amount invested at 5%
Let y = amount invested at 7%
(1) The total invested is $2000: x+y = 2000
(2) The total interest is $108: .05x+.07y = 108
One solution method is to solve the first equation for either x or y and substitute in the second equation:
y = 2000-x
.05x+.07(2000-x) = 108
That gives you a single equation in a single variable, like the one shown in the response from the other tutor.
The other standard algebraic method for solving a system of two equations is elimination. Here is one possible way to do that
x+y = 2000 (the first equation, as it is)
x+1.4y = 2160 (the second equation, multiplied by 20 -- because 20*(.05) = 1)
0.4y = 160
y = 160/0.4 = 400
y = $400 was invested at 7%, so $2000-$400 = $1600 was invested at 5%
ANSWER: $1600 at bank A, $400 at bank B
NOTE: Solving a problem with formal algebra using a single variable (and therefore a single equation) almost always makes the actual solution easier and faster. However, a beginning algebra student should understand how to set up and solve the problem using two variables, as this assignment asks you to do.
Second NOTE: If formal algebra is not required, here is a quick and easy way to solve this kind of "2-part mixture" problem.
(1) If all $2000 were invested at 5%, the interest would be $100; if all were invested at 7%, the interest would be $140.
(2) The actual interest, $108, is $8/$40 = 1/5 of the way from $100 to $140.
(3) That means 1/5 of the total was invested at the higher rate.
ANSWER: 1/5 of $2000, or $400, was invested at 7%; the other $1600 at 5%.
Answer by ikleyn(52770)  (Show Source):
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