Solving Linear Equations

Introduction to Linear Equations

Linear equations are fundamental in algebra and represent statements where two algebraic expressions are equal. The solution to a linear equation is the value of the variable that makes the equation true when substituted.

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

• Linear Equation in One Variable: An equation of the form , where , , and are real numbers and is the variable.

• Solution Set: The set of values that satisfy the equation, written in set brackets { }.

• Example: Determine whether a given value is a solution to the equation .

Identifying Linear Equations

Not all equations are linear. Linear equations have variables raised only to the first power and do not involve products of variables or higher powers.

• Linear Equation:

• Nonlinear Equation: or

• Example: is linear; is not.

Expressions vs. Equations

It is important to distinguish between expressions and equations. An expression does not have an equal sign, while an equation does.

• Expression:

• Equation:

Solving Linear Equations

Solving linear equations involves isolating the variable using properties of equality. The main properties are addition, subtraction, multiplication, and division.

• Addition Property: If , then

• Subtraction Property: If , then

• Multiplication Property: If , then

• Division Property: If , then (for )

• Example: Solve

Checking Solutions

Always check your solution by substituting the value back into the original equation to verify it makes the statement true.

• Example: For , is a solution because .

Solving Equations with Fractions and Decimals

When equations involve fractions or decimals, clear them by multiplying both sides by the least common denominator (LCD) or a power of 10.

• Example:

• Example:

Summary Table: Properties of Equality

The properties of equality are essential tools for solving equations. The table below summarizes their use:

Solving Linear Equations: Step-by-Step

Follow these steps to solve any linear equation:

1. Simplify both sides of the equation (distribute, combine like terms).

2. Use addition/subtraction properties to collect variable terms on one side and constant terms on the other.

3. Use multiplication/division properties to isolate the variable.

4. Check your solution by substituting back into the original equation.

Examples of Linear Equations

Practice Problems

• Solve

• Solve

• Solve ;

• Solve ;

Additional info:

These notes cover the foundational concepts and procedures for solving linear equations, which are essential for intermediate algebra students. The included images visually reinforce the algebraic concepts and steps described.