earned on these investments was 3890 dollars. How much money did he invest at each rate?

In order to solve this problem, one must construct a system of equations. Let x represent the amount of money invested at 7% and y the amount invested at 5%. In order to find the amount of interest, we construct this equation:

3,890 = .07x + .05y

Remember that 7% is .07 when expressed as a decimal (and similarly for 5%). 3,890 is the amount of interest earned in one year. .07x is 7% of x and .05y is 5% of y, which are the amounts of interest each investment earned in one year. We add them together to get the total interest earned.

Now we need to express what x and y are in a separate equation. We know that x + y must be $63,000 because x and y are parts of the total investment, which was $63,000. That equation looks like this:

63,000 = x + y

From here, we go about solving our system of equations. First, let's find a value for x:

63,000 = x + y

63,000 - y = x

Now that we have a value for x, we can substitute it into the other equation and solve for y:

3,890 = .07x + .05y

3,890 = .07(63,000 - y) + .05y

3,890 = (4,410 - .07y) + .05y

3,890 = 4,410 - .02y

-520 = -.02y

26,000 = y

At this point we know that $26,000 was invested at 5%. If we take 5% of $26,000 we get $1,390 in interest for this portion of the total investment. Since $1,390 is less than the total interest earned of $3,890 we can move on relatively confident that we are on the right track.

The final step to solving this problem is to follow the same pattern, but to solve for y in the first step and substitute it in to find the value of x.

Solve for y:

63,000 = x + y

63,000 - x = y

Now substitute that in and find the value of x:

3,890 = .07x + .05(63,000 - x)

3,890 = .07x + (3,150 - .05x)

3,890 = .02x + 3,150

740 = .02x

37,000 = x

We find that $37,000 was invested at 7%. This investment yields $2,590 in interest. Remember, the problem isn't done until the answer is checked. For a system of equations, one must check both equations.

First the equation of investments:

63,000 = x + y

63,000 = 37,000 + 23,000

63,000 = 63,000

Then the equation of interest:

3,890 = .07x + .05y

3,890 = .07(37,000) + .05(26,000)

3,890 = 2,590 + 1,300

3,890 = 3,890

Both equations check out and we are finished!

## Comments

Thank you very much!

You're very welcome!