A **linear equation** is an equation of a straight line, written in one variable. The only power of the variable is 1. Linear equations in one variable may take the form [latex]ax+b=0[/latex] and are solved using basic algebraic operations.

We begin by classifying linear equations in one variable as one of three types: identity, conditional, or inconsistent. An **identity equation** is true for all values of the variable. Here is an example of an identity equation.

The **solution set** consists of all values that make the equation true. For this equation, the solution set is all real numbers because any real number substituted for [latex]x[/latex] will make the equation true.

A **conditional equation** is true for only some values of the variable. For example, if we are to solve the equation [latex]5x+2=3x - 6[/latex], we have the following:

The solution set consists of one number: [latex]\{-4\}[/latex]. It is the only solution and, therefore, we have solved a conditional equation.

An **inconsistent equation** results in a false statement. For example, if we are to solve [latex]5x - 15=5\left(x - 4\right)[/latex], we have the following:

Indeed, [latex]-15\ne -20[/latex]. There is no solution because this is an inconsistent equation.

Solving linear equations in one variable involves the fundamental properties of equality and basic algebraic operations. A brief review of those operations follows.

### A General Note: Linear Equation in One Variable

A linear equation in one variable can be written in the form

where *a* and *b *are real numbers, [latex]a\ne 0[/latex].

### How To: Given a linear equation in one variable, use algebra to solve it.

The following steps are used to manipulate an equation and isolate the unknown variable, so that the last line reads x=_________, if *x *is the unknown. There is no set order, as the steps used depend on what is given:

- We may add, subtract, multiply, or divide an equation by a number or an expression as long as we do the same thing to both sides of the equal sign. Note that we cannot divide by zero.
- Apply the distributive property as needed: [latex]a\left(b+c\right)=ab+ac[/latex].
- Isolate the variable on one side of the equation.
- When the variable is multiplied by a coefficient in the final stage, multiply both sides of the equation by the reciprocal of the coefficient.

### Example 1: Solving an Equation in One Variable

Solve the following equation: [latex]2x+7=19[/latex].

### Solution

This equation can be written in the form [latex]ax+b=0[/latex] by subtracting [latex]19[/latex] from both sides. However, we may proceed to solve the equation in its original form by performing algebraic operations.

The solution is [latex]x=6[/latex].

### Example 2: Solving an Equation Algebraically When the Variable Appears on Both Sides

Solve the following equation: [latex]4\left(x - 3\right)+12=15 - 5\left(x+6\right)[/latex].

### Solution

Apply standard algebraic properties.

### Analysis of the Solution

This problem requires the distributive property to be applied twice, and then the properties of algebra are used to reach the final line, [latex]x=-\frac{5}{3}[/latex].