Given this matrix, perform the RREF operation on a calculator to find the solutions to the nearest hundredth.
$\left[\begin{array}{ccc} 100 & -1 & -200 \ 75 & 1 & 2,000 \end{array}\right]$
In $\square$ months, Rafiki will be able to pay off the remaining balance of $\$\square$
Answer (2)
Set up the system of equations from the given matrix: 100 x − y = − 200 and 75 x + y = 2000 .
Perform the RREF operation to find the values of x and y .
Round the values of x and y to the nearest hundredth: x ≈ 10.29 and y ≈ 1228.57 .
Interpret the results: In approximately 10.29 months, Rafiki will have a remaining balance of 1228.57 , so t h e f ina l an s w er i s \boxed{10.29} an d \boxed{ 1228.57} .
Explanation
Understanding the Problem We are given a matrix and asked to perform the Reduced Row Echelon Form (RREF) operation to find the solution to the system of equations represented by the matrix. The matrix is: [ 100 − 1 − 200 75 1 2 , 000 ] The RREF operation will help us solve for the unknowns, which represent the number of months and the remaining balance.
Setting up the Equations The given matrix represents the following system of equations: 100 x − y = − 200 75 x + y = 2000 where x represents the number of months and y represents the remaining balance. We need to solve this system for x and y .
Performing RREF After performing the RREF operation (using a calculator or software), we obtain the values for x and y . The result of the RREF operation is: x ≈ 10.285714285714286 y ≈ 1228.5714285714287
Rounding the Results We are asked to round the answers to the nearest hundredth. Therefore, x ≈ 10.29 y ≈ 1228.57
Interpreting the Results The value of x represents the number of months, and the value of y represents the remaining balance. Therefore, in approximately 10.29 months, Rafiki will be able to pay off the recurring expenses, leaving a balance of approximately $1228.57.
Final Answer Therefore, the final answer is: In 10.29 months, Rafiki will be able to pay off the recurring expenses, leaving a balance of $\boxed{ 1228.57} .
Examples
Understanding how to balance income and expenses is crucial in personal finance. This problem demonstrates how to determine the time it takes to pay off recurring expenses and the remaining balance. For example, if Rafiki has a debt and makes regular payments, this calculation helps him estimate when he will pay off the debt and how much he will still owe at that time. This is essential for budgeting and financial planning.
Using RREF on the matrix provides solutions for months and balance: approximately 10.29 months for repayment and a remaining balance of about $1228.57. Rafiki will pay off expenses in this timeframe. The final results are 10.29 months and $1228.57.
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