A sum of 10,000 is invested in three accounts that pay 6 %,...

$A$ sum of $\$ 10,000$ is invested in three accounts that pay $6 \%, 8 \%,$ and $10 \%$ interest. Twice as much money is invested in the account paying $10 \%$ as in the account paying $6 \%,$ and the total annual interest from all three accounts is $\$ 842$. (a) Write a system of equations whose solution gives the amount invested in each account. Be sure to state what each variable represents. (b) Solve the system of equations.

$A$ sum of $\$ 10,000$ is invested in three accounts that pay $6 \%, 8 \%,$ and $10 \%$ interest. Twice as much money is invested in the account paying $10 \%$ as in the account paying $6 \%,$ and the total annual interest from all three accounts is $\$ 842$.
(a) Write a system of equations whose solution gives the amount invested in each account. Be sure to state what each variable represents.
(b) Solve the system of equations.

Key Concepts

Methods for Solving Systems (Substitution and Elimination)

These techniques are used to find the solution to a system of equations. The substitution method involves solving one equation for one variable and replacing this expression in another equation, while the elimination method involves adding or subtracting equations to eliminate a variable. Both methods are crucial for efficiently solving simultaneous equations in linear algebra.

Variable Representation

Variable representation is the strategy of assigning symbols to unknown quantities. This abstraction allows one to establish general relationships and form equations that can be manipulated algebraically, providing a clear method to denote and solve for unknown values.

Financial Mathematics (Simple Interest)

This concept refers to the calculation of interest on a principal amount, often in relation to a specific percentage rate over a period. Understanding how to compute simple interest is essential for creating equations that model financial scenarios, especially those involving investments or loans.

Word Problem Translation

This involves interpreting a real-world scenario and expressing it mathematically. The process requires identifying relationships described in the problem, formulating equations based on these relationships, and organizing information in a way that enables the application of algebraic methods to find a solution.

System of Linear Equations

This concept involves using a set of two or more linear equations in two or more variables. In such systems, the solution is the set of values that simultaneously satisfy all the equations, making it a fundamental tool in algebra for dealing with multiple constraints or conditions at the same time.

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