Using a Matrix to Solve an Investment Problem

A manufacturing firm borrowed $30,000.  Part of the money was borrowed at 12%, part at 14%, and part at 16%.  The annual simple interest was $4280.  The total amount borrowed at 12% and 14% was twice the amount borrowed at 16%.

Let x represent the amount borrowed at 12%, y represent the amount borrowed at 14%, and z represent the amount borrowed at 16%.  Use the given information to write 3 equations using x, y, and z.

   

Write the 3 equations in standard form and translate the system of equations into a matrix.

       

Use matrix row operations to transform the matrix from question 2 into an equivalent matrix with 1 zero in the left column of both the second and the third rows, and an additional zero in the second column of the third row.

    

Translate the last row of the matrix from question 3 into an equation, and solve this equation for z.

 

Translate the second row of the matrix from question 3 into an equation, and solve this equation for y by substituting the value for z found in question 4.

     

 Translate the first row of the matrix from question 3 into an equation, and solve this equation for x by substituting the values for y and z found in questions 4 and 5.

    

State the amounts borrowed at each rate using the answers from questions 4, 5, and 6.