Overview
This lecture reviews strategies and examples for solving application problems using linear equations, focusing on mixture, investment, and other real-life scenarios, and ends with an optional quiz.
Steps for Solving Application Problems
- Read the problem carefully to identify what is given and what needs to be found.
- Assign a variable (e.g., x, t, n) to the unknown value.
- Write an equation relating the unknown to the given information.
- Solve the equation using algebraic methods.
- State the answer and check for reasonableness.
- Verify the solution is correct.
Mixture Problems Example
- Example: Find gallons of 25% antifreeze needed to mix with 5 gallons of 10% solution to get 15% solution.
- Use a table: list strengths, gallons, and pure antifreeze amounts.
- Multiply percentages and quantities across rows; add values in columns.
- Set total pure antifreeze before and after mixing equal: 0.25x + 0.5 = 0.15(x + 5).
- Solve resulting equation for x to find the number of gallons needed.
Investment Problems Example
- Example: Ryan earned $784 from two investments totaling $28,000 at 2.4% and 3.1%.
- Assign x to the amount at 2.4%; the rest is 28,000 - x at 3.1%.
- Multiply each amount by its rate and sum to get total interest: 0.024x + 0.031(28,000 - x) = 784.
- Solve for x to find each investment amount.
Solving Given Equations
- Substitute known values into the given formula.
- Example: Find flow F for a given percent p using p = 1.06F + 7.18.
- Rearrange and solve for the unknown variable.
Linear Modeling Example
- Given linear model for US health expenditures: y = 343x + 4512, x = years after 2000.
- For a specific year, substitute x and solve for y.
- For a target expenditure, set y and solve for x, then interpret as a calendar year.
Key Terms & Definitions
- Variable — a symbol, often a letter, representing an unknown value.
- Linear Equation — an equation of the form ax + b = c, where the solution is found by isolating x.
- Mixture Problem — a problem involving combining solutions of different strengths.
- Modeling — using equations to represent real-world scenarios.
Action Items / Next Steps
- Optional quiz: Find rectangle dimensions given perimeter and relationship between length and width; submit in Moodle or by email.
- Review the steps for setting up equations from word problems for future assignments.