SOLUTION: To buy both a new car and a new house, Tina sought two loans totalling $78,825. The simple interest rate on the first loan was 0.2%, while the simple interest rate on the second lo

Question 354768: To buy both a new car and a new house, Tina sought two loans totalling $78,825. The simple interest rate on the first loan was 0.2%, while the simple interest rate on the second loan was 5.0%. At the end of the first year, Tina paid a combined interest payment of $2817.23. What were the amounts of the two loans?
So far I have this but don't know where to go from there:
x + y = 78,825
0.02x + 0.05y = 2817.23
Found 2 solutions by stanbon, solver91311:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
To buy both a new car and a new house, Tina sought two loans totalling $78,825. The simple interest rate on the first loan was 0.2%, while the simple interest rate on the second loan was 5.0%. At the end of the first year, Tina paid a combined interest payment of $2817.23. What were the amounts of the two loans?
So far I have this but don't know where to go from there:
x + y = 78,825
0.02x + 0.05y = 2817.23
----
Multiply thru the 1st equation by 2.
Multiply thru the 2nd equation by 100.
---------------
2x + 2y = 2*78,825
2x + 5y = 281723
---------------------------
Subtract the top equation from the bottom and solve for "y":
3y = 124073
y = $41357.67
----
x = 78825-41357.67
x = $37467.33
===================
Cheers,
Stan H.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

Typo or error? Your problem statement says the first loan is at 0.2% simple interest, which really looks like a typo, but if 0.2% is actually correct, then your second equation has an error and should be:

Either way, there are three basic ways to go about solving your system:

1. Substitution:

Solve for either of the variables in terms of the other. Let's solve for :

Now substitute into the 2nd equation. (I'm going to go with my gut feeling and say you had a typo in the problem statement, but you can make the adjustment if necessary)

And then just solve for .

2. Elimination:

Multiply your second equation by -50:

Then add the two equations:

Notice that goes away leaving you with an equation in

3. Cramer's Rule:

Create the coefficient matrix of your system:

And then calculate the determinant, , of that matrix.

The determinant

So you will have

Then replace the first column of your matrix with the constant values from your equations:

and then using the same process, calculate the determinant,

Next, replace the second column:

From which you calculate

Finally,

and

John

My calculator said it, I believe it, that settles it

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