Linear Equations and Inequalities in One Variable

Definition of a Linear Equation in One Variable

A linear equation in one variable is an equation that can be written in the standard form:

where a, b, and c are real numbers, and a \neq 0. The variable x appears only to the first power and is not multiplied by itself or any other variable.

• Example: is a linear equation in one variable.

The Addition Property of Equality

The Addition Property of Equality states that the same real number (or algebraic expression) may be added to both sides of an equation without changing the equation’s solution. Symbolically:

If , then .

• This property is fundamental for solving equations, as it allows us to isolate variables by adding or subtracting terms on both sides.

Solving Equations Using the Addition Property

• Step 1: Identify the variable to isolate.

• Step 2: Add or subtract the same value from both sides to move constants or variable terms as needed.

• Step 3: Simplify both sides to solve for the variable.

Example 1: Solving an Equation Using the Addition Property

Solve .

Add 9 to both sides:

The solution set is .

Subtracting from Both Sides of an Equation

Subtraction is the addition of the additive inverse. Thus, the addition property also allows us to subtract the same number from both sides without changing the solution.

Example 2: Subtracting the Same Number from Both Sides

Solve and check: .

Check: (True). The solution set is .

Example 3: Isolating the Variable on the Right

When the variable appears on the right side, use the same addition or subtraction steps to isolate it. (Original content incomplete; see "Additional info" below.)

Additional info: In such cases, rearrange the equation so the variable is isolated, then solve as above.

Example 4: Combining Like Terms before Using the Addition Property

Solve and check: .

Check: (True). The solution set is .

Example 5: Using the Addition Property to Isolate Variable Terms

Solve and check: .

Check: (True). The solution set is .

Example 6: Solving an Equation by Isolating the Variable

Solve and check: .

Check: (True). The solution set is .

Solving Applied Problems Using Formulas

Many real-world problems can be modeled with linear equations. To solve these, substitute known values into the formula and solve for the unknown variable.

Example 7: Application

There is a relationship between the number of words in a child’s vocabulary, , and the child’s age, , in months, for ages between 15 and 50 months, inclusive. The formula is:

Find the number of words in a child’s vocabulary at the age of 50 months.

At the age of 50 months, a child has a vocabulary of 2100 words.