Equations in One Variable

Linear Equations in One Variable

A linear equation in one variable is an equation that can be expressed in the form , where a and b are integers, and x is a variable with only one solution.

• Standard Form:

• Solution: Isolate x to find its value.

Properties of Equality

These properties allow us to manipulate equations to solve for the variable:

Example

• Solve Solution:

Types of Solutions for Linear Equations

Linear equations can have different types of solutions:

1. Conditional Equation: True for some value(s) of the variable, but not all.

2. Inconsistent Equation: No solution; equations that are impossible to solve.

3. Identity Equation: True for every value of the variable.

Equations Involving Rational Expressions

Rational equations contain fractions with variables in the denominator. To solve:

• Multiply both sides by the least common denominator (LCD) to eliminate fractions.

Example

• Solve Solution: Multiply both sides by and solve for .

Equations Involving Absolute Value

Absolute value equations require isolating the absolute value before solving:

Example

• Solve Solution: or

Word Problems and Applications

Linear equations are used to solve real-world problems, such as:

• Finding the cost before tax: If the total cost is and the tax rate is , then .

• Dividing lengths: If a piece is split into parts with relationships, set up an equation to solve for each part.

Example

• A carpenter used 45 ft in three pieces to trim a garage door. If the long piece was 5 ft longer than twice the length of each shorter piece, let be the length of a short piece. Then .

Additional info:

• When solving equations with fractions, always multiply by the LCD to clear denominators.

• For absolute value equations, check for extraneous solutions by substituting back into the original equation.