뒤로Precalculus Study Notes: Linear Equations, Applications, and Literal Equations
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Models and Applications
Algebraic Expressions and Formulas
Understanding algebraic expressions and formulas is fundamental in solving real-world problems using mathematics. An algebraic expression is a mathematical phrase that combines numbers, variables, and operation symbols, but does not contain an equals sign or inequality symbol. A formula is an equation that expresses a relationship between variables.
Simple Interest Formula: Where I is interest, P is principal, r is rate, and t is time.
Perimeter of a Rectangle: Where L is length and W is width.
Strategy for Solving Word Problems
Solving word problems involves translating real-world situations into mathematical equations. The following steps provide a systematic approach:
Read the problem carefully and identify what is given and what is required.
Let a variable (e.g., x) represent an unknown quantity.
Express other unknowns in terms of x.
Write an equation that models the problem.
Solve the equation and check your solution.
Examples of Algebraic Expressions
Example 1: "Multiply a number by 8. Add 11 to this product. Subtract this sum from the number." Algebraic expression: Simplified:
Example 2: "When one-fifth of a number is added to one-fourth of the number, the sum is 27." Equation:
Solving Formulas for a Variable
Isolating a Variable
Sometimes, a formula contains the same variable more than once. To solve for a specific variable, collect all terms with that variable on one side and factor as needed.
Example: To solve for P:
Factor P:
Isolate P:
Application Problems
Linear Models in Real Life
Many real-world scenarios can be modeled using linear equations. Below are several examples:
Example 1: Depreciation of a Car A car worth $26,000 depreciates by $2,000 per year. The value after x years is: To find when the value is years
Example 2: Comparing Health Club Memberships Club A: Club B: Set to find when costs are equal: months Cost:
Example 3: Electronic Toll Pass Regular toll: Pass: Set equal: times
Graphical Representation of Linear Models
Graphs are useful for visualizing linear relationships. For the car depreciation example, the correct graph should have:
Vertical axis (y) representing car value ($)
Horizontal axis (x) representing years
A line starting at when and decreasing by per year
Literal Equations
Solving for a Specified Variable
Literal equations involve solving for one variable in terms of others. This is useful in rearranging formulas for different applications.
Example 1: for
Example 2: for
Example 3: for
Example 4: for
Example 5: for
Example 6: for
Additional Application Examples
Mixture and Investment Problems: Use systems of equations to solve for unknowns, such as amounts invested at different rates or proportions in mixtures.
Percent Problems: For price reductions, let be the original price. After a 90% reduction, .
Geometry Applications: For rectangles, use perimeter and relationships between length and width to set up equations.
