Linear Equations

Definition and Key Concepts

A linear equation is a statement that two algebraic expressions are equal. Linear equations are fundamental in algebra and precalculus, forming the basis for solving many types of mathematical problems.

• Equation: A statement that two expressions are equal. Example:

• Solve: To find all numbers that make the equation a true statement.

• Solution: A number that makes the equation a true statement.

• Solution Set: All the numbers that make the equation a true statement.

General Form of a Linear Equation

Any linear equation in one variable can be written in the form:

• , where

Examples

• Example 1:

• Example 2:

Steps for Solving Linear Equations

Systematic Approach

To solve a linear equation, follow these steps:

1. List any restrictions on the variable. For example, denominators cannot be zero.

2. If necessary, clear the equation of fractions by multiplying both sides by the Least Common Multiple (LCM) of all denominators.

3. Remove all parentheses and simplify.

4. Collect all terms containing the variable on one side and all remaining terms on the other side.

5. Simplify and solve for the variable.

6. Check your solution(s) by substituting back into the original equation.

Solving Equations with Fractions

Clearing Fractions

When an equation contains fractions, it is often helpful to eliminate them by multiplying both sides by the Least Common Denominator (LCD) of all fractions present.

• Step: Multiply both sides by the LCD to clear all fractions.

Example:

Equations with No Solution

Identifying Inconsistent Equations

Some equations have no solution, meaning there is no value for the variable that makes the equation true.

• Example:

In this example, simplifying both sides may lead to a contradiction, such as , indicating no solution exists.

Word Problems Involving Linear Equations

General Strategy for Solving Word Problems

Word problems require translating real-world situations into mathematical equations. The following steps provide a systematic approach:

1. Read the problem carefully to understand what is being asked and identify realistic possibilities for the answer.

2. Assign a variable to represent the unknown quantity, and express other unknowns in terms of this variable if necessary.

3. List all known facts and translate them into mathematical expressions or equations. Diagrams or tables may help organize information.

4. Solve the equation for the variable, and answer the question in a complete sentence.

5. Check your answer against the facts in the problem to ensure it is reasonable.

Example: Wage Calculation

Problem: Andy grossed $435 in one week by working 52 hours. His employer pays time-and-a-half for all hours worked in excess of 40 hours. What is Andy’s hourly wage?

• Let = Andy's hourly wage.

• Regular hours: 40 hours at per hour.

• Overtime hours: hours at per hour.

• Total earnings:

• Simplify:

• Answer: Andy’s hourly wage is $7.50 per hour.

Summary Table: Steps for Solving Linear Equations

Additional info: The notes above are foundational for Precalculus students, covering the essential process of solving linear equations, handling equations with fractions, recognizing equations with no solution, and applying these skills to word problems.