SOLUTION: $5,000 is distributed among three investment types: at 8%, 3%, and 1%. the total of amounts invested at 8% and at 1% is equal the amount invested at 3%. the total interest after on

Algebra ->  Matrices-and-determiminant -> SOLUTION: $5,000 is distributed among three investment types: at 8%, 3%, and 1%. the total of amounts invested at 8% and at 1% is equal the amount invested at 3%. the total interest after on      Log On


   



Question 867083: $5,000 is distributed among three investment types: at 8%, 3%, and 1%. the total of amounts invested at 8% and at 1% is equal the amount invested at 3%. the total interest after one year from all three accounts is $144. How much was invested at each account? Solve using matrices.
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
a+b+c=5000
.01a+.03b+.08c=144
.01a-.03b+.08c=0
Add (-1/100 * row1) to row2
1 1 1 5000
0 1/50 7/100 94
1/100 -3/100 8/100 0
Add (-1/100 * row1) to row3
1 1 1 5000
0 1/50 7/100 94
0 -1/25 7/100 -50
Divide row2 by 1/50
1 1 1 5000
0 1 7/2 4700
0 -1/25 7/100 -50
Add (1/25 * row2) to row3
1 1 1 5000
0 1 7/2 4700
0 0 21/100 138
Divide row3 by 21/100
1 1 1 5000
0 1 7/2 4700
0 0 1 4600/7
Add (-7/2 * row3) to row2
1 1 1 5000
0 1 0 2400
0 0 1 4600/7
Add (-1 * row3) to row1
1 1 0 30400/7
0 1 0 2400
0 0 1 4600/7
Add (-1 * row2) to row1
1 0 0 13600/7
0 1 0 2400
0 0 1 4600/7

Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 3 variables







First let A=%28matrix%283%2C3%2C1%2C1%2C1%2C0.01%2C0.03%2C0.08%2C0.01%2C-0.03%2C0.08%29%29. This is the matrix formed by the coefficients of the given system of equations.


Take note that the right hand values of the system are 5000, 144, and 0 and they are highlighted here:




These values are important as they will be used to replace the columns of the matrix A.




Now let's calculate the the determinant of the matrix A to get abs%28A%29=0.0042. To save space, I'm not showing the calculations for the determinant. However, if you need help with calculating the determinant of the matrix A, check out this solver.



Notation note: abs%28A%29 denotes the determinant of the matrix A.



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Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5Bx%5D (since we're replacing the 'x' column so to speak).






Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=8.16. Again, as a space saver, I didn't include the calculations of the determinant. Check out this solver to see how to find this determinant.



To find the first solution, simply divide the determinant of A%5Bx%5D by the determinant of A to get: x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%288.16%29%2F%280.0042%29=1942.85714285714



So the first solution is x=1942.85714285714




---------------------------------------------------------


We'll follow the same basic idea to find the other two solutions. Let's reset by letting A=%28matrix%283%2C3%2C1%2C1%2C1%2C0.01%2C0.03%2C0.08%2C0.01%2C-0.03%2C0.08%29%29 again (this is the coefficient matrix).




Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5By%5D (since we're replacing the 'y' column in a way).






Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=10.08.



To find the second solution, divide the determinant of A%5By%5D by the determinant of A to get: y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%2810.08%29%2F%280.0042%29=2400



So the second solution is y=2400




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Let's reset again by letting A=%28matrix%283%2C3%2C1%2C1%2C1%2C0.01%2C0.03%2C0.08%2C0.01%2C-0.03%2C0.08%29%29 which is the coefficient matrix.



Replace the third column of A (that corresponds to the variable 'z') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5Bz%5D






Now compute the determinant of A%5Bz%5D to get abs%28A%5Bz%5D%29=2.76.



To find the third solution, divide the determinant of A%5Bz%5D by the determinant of A to get: z=%28abs%28A%5Bz%5D%29%29%2F%28abs%28A%29%29=%282.76%29%2F%280.0042%29=657.142857142857



So the third solution is z=657.142857142857




====================================================================================

Final Answer:




So the three solutions are x=1942.85714285714, y=2400, and z=657.142857142857 giving the ordered triple (1942.85714285714, 2400, 657.142857142857)




Note: there is a lot of work that is hidden in finding the determinants. Take a look at this 3x3 Determinant Solver to see how to get each determinant.