I have a math question I am stuck on. It is on the applications of linear systems. This is the question: A club raised 5000$ for a trip, they invested part of the money into a savings account with 4% interest, they put the rest of the money into a government bond with 3.5% interest. Overall they earned 190$ in interest in a year. How much was invested at each rate?

Question

I have a math question I am stuck on. It is on the applications of linear systems. This is the question: A club raised 5000$ for a trip, they invested part of the money into a savings account with 4% interest, they put the rest of the money into a government bond with 3.5% interest. Overall they earned 190$ in interest in a year. How much was invested at each rate?

I proceeded to create two equations, I made 2 variables which are x and y and these represent the unknown variables. The 2 equations I made are as followed:
x + y = 5000
0.04x + 0.035y = 190

I decided to use elimination to solve this problem so I multiplied the first equation by 3.5 and then I multiplied the bottom equation by 100 then the result is this:

3.5x + 3.5y = 17500
4x + 3.5y = 19000

I subtracted these two equations to get:
- 0.5x + 0y = -1500

So I divided both - 0.5x and -1500 by - 0.5 and got
x = 30000

But when I substitute 30000 for x in the first equation to find y I get -25000. I clearly did something wrong because the solutions for x and y don't add up to 5000. Could someone please help me understand where I went wrong and help give me the knowledge and tools to answer this question? It would be extremely appreciated!

Answer 1

misplaced decimal point in the last step

-1500 / -0.5 = 3000

Answer 2

When you divide 1500 by 0.5 you get 3000, not 30000

so in x+y = 5000, y = 2000

For your equations, I would have used substitution, using y = 5000-x from the first
then in the 2nd:
.03x + .035(5000-x) = 190
etc

Answer 3

Thank you so much R.Scott!! Very much appreciated, you've made my day.

Answer 4

Don't worry, I can help you understand where you went wrong and guide you through solving this problem correctly.

You're on the right track with creating your two equations. Let's review them:

Equation 1: x + y = 5000 (representing the total amount invested)
Equation 2: 0.04x + 0.035y = 190 (representing the total interest earned)

To solve this system of equations, you can use the method of substitution. Here's how to proceed:

Step 1: Solve Equation 1 for one variable in terms of the other variable. Let's solve for x:
x = 5000 - y

Step 2: Substitute the expression for x into Equation 2:
0.04(5000 - y) + 0.035y = 190

Step 3: Simplify and solve for y:
200 - 0.04y + 0.035y = 190
0.005y = -10
y = -10 / 0.005
y = 2000

Step 4: Substitute the value of y back into the first equation to solve for x:
x + 2000 = 5000
x = 5000 - 2000
x = 3000

So, the correct solution is x = 3000 and y = 2000. This means $3000 were invested at 4% interest, and $2000 were invested at 3.5% interest.

Now let's discuss where you went wrong in your initial attempt:

You made an error in your elimination step. When you multiplied the first equation by 3.5, the correct resulting equation should have been 3.5x + 3.5y = 17500. Similarly, when you multiplied the second equation by 100, it should have been 4x + 3.5y = 19000. With these corrected equations, you would follow the same steps as mentioned earlier to get the correct answer.

I hope this explanation helps you understand the mistake and provides you with a clear path to solving the problem. Let me know if you have any further questions!