You invested \$20,000 in two accounts paying 1.45% and 1.59% annual interest. If the total interest earned for the year was \$307.50, how much was invested at each rate?

Verified step by step guidance

1. Define variables to represent the amounts invested: let \(x\) be the amount invested at 1.45%, and \(y\) be the amount invested at 1.59%.
2. Write an equation representing the total amount invested: \(x + y = 20000\).
3. Write an equation representing the total interest earned: \(0.0145x + 0.0159y = 307.50\).
4. Use the first equation to express one variable in terms of the other, for example, \(y = 20000 - x\).
5. Substitute this expression for \(y\) into the interest equation and solve for \(x\), then use the value of \(x\) to find \(y\).

Key Concepts

Systems of Linear Equations

A system of linear equations involves two or more equations with multiple variables. Solving such systems helps find the values of unknowns that satisfy all equations simultaneously. In this problem, two variables represent amounts invested at different rates, and their relationships form a system to solve.

Interest Calculation

Interest is the amount earned or paid on a principal over time, often expressed as a percentage rate. Simple interest is calculated by multiplying the principal by the interest rate and time. Understanding how to compute interest is essential to relate the invested amounts to the total interest earned.

Setting Up Equations from Word Problems

Translating a word problem into mathematical equations involves identifying known values, unknowns, and their relationships. This skill is crucial to form equations that model the problem accurately, enabling the use of algebraic methods to find solutions.