뒤로Solving Linear Equations and Applications in Precalculus
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Equations and Inequalities
Problem Solving with Linear Equations
Linear equations are fundamental tools for modeling and solving real-world problems in algebra and precalculus. The process of solving word problems using linear equations involves a systematic approach:
Step 1: Let x represent one of the unknown quantities.
Step 2: Express other unknowns in terms of x.
Step 3: Write an equation in x that models the problem's conditions.
Step 4: Solve the equation and answer the question.
Step 5: Check the solution in the original context of the problem.
Example 1: Median Starting Salaries
This example demonstrates how to set up and solve a linear equation based on relationships between unknowns.
Given: The median starting salary of a computer science major exceeds that of an education major by $21,000. The median starting salary of an economics major exceeds that of an education major by $14,000. Combined, their median starting salaries are $140,000.
Let x = median starting salary of an education major.
Then, computer science: x + 21; economics: x + 14.
Equation:
Solution: (education), (computer science), (economics).
Check: (verifies the solution).
Example 2: Comparing Texting Plans
This example illustrates how to model a cost comparison problem with linear equations.
Plan A:
Plan B:
Equation:
Solution: (the number of texts where costs are equal).
Check: Both plans cost $39 for 300 texts.
Example 3: Price Reduction Problem
This example shows how to find the original price before a percentage reduction using a linear equation.
Let x = original price.
After 30% reduction:
Equation:
Solution: (original price).
Check: (verifies the solution).
Example 4: Investment Allocation
This example involves dividing an inheritance between two investments with different interest rates to achieve a specified total interest.
Let x = amount invested at 9%.
Amount at 11%:
Equation:
Solution: at 9%, at 11%.
Check: (verifies the solution).
Example 5: Basketball Court Dimensions
This example demonstrates how to use perimeter formulas and linear equations to find the dimensions of a rectangle given a relationship between length and width.
Let x = width; length =
Perimeter:
Solution: Width = 50 ft, Length = 94 ft.
Check: (verifies the solution).
Solving a Formula for a Variable
Isolating a Variable in a Formula
Solving a formula for a variable means rewriting the equation so that the chosen variable is alone on one side. This is essential for rearranging equations in algebra and applied mathematics.
Example: Solve for .
Solution:
Solving a Formula for a Variable That Occurs Twice
Sometimes, the variable to be isolated appears in more than one term. Factorization and algebraic manipulation are used to solve for the variable.
Example: Solve for .
Solution:
