The given amount of annual interest is earned from a total...

The given amount of annual interest is earned from a total of $\$ 12,000$ invested in two funds paying simple interest. Write and solve a system of equations to find the amount invested at each given rate. $$\begin{array}{lll}\text { Annual Interest } & \text { Rate } \mathbf{1} & \text { Rate } \mathbf{2} \\\text { S630 } & 4 \% & 6 \% \end{array}$$

The given amount of annual interest is earned from a total of $\$ 12,000$ invested in two funds paying simple interest. Write and solve a system of equations to find the amount invested at each given rate.

$$\begin{array}{lll}\text { Annual Interest } & \text { Rate } \mathbf{1} & \text { Rate } \mathbf{2} \\\text { S630 } & 4 \% & 6 \% \end{array}$$

Key Concepts

Linear Equations

Linear equations are algebraic expressions that involve variables raised only to the first power. In investment problems, each condition (like the total amount invested or the total interest earned) can be modeled using a linear equation, which is then solved by methods such as substitution or elimination within the system of equations.

Variable Representation

Variable representation involves assigning symbols—typically letters—to unknown quantities, such as the amounts invested at different interest rates. This abstraction helps in formulating equations that represent the underlying relationships in the problem, making it easier to apply algebraic methods to solve for the variables.

Simple Interest

Simple interest is a method for calculating the interest charge on a principal amount where the interest is computed solely on the original sum, without compounding over time. It is typically expressed as a percentage of the principal per time period, and the formula I = P × r is used, where I is the interest earned, P is the principal amount, and r is the interest rate.

System of Equations

A system of equations is a collection of two or more equations with a common set of variables. In financial problems, these systems are constructed to represent different conditions, such as the total investment amount and the overall interest earned, allowing one to determine the values of the unknown variables simultaneously.

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