2.2 Linear equations in one variable

Caroline is a full-time college student planning a spring break vacation. To earn enough money for the trip, she has taken a part-time job at the local bank that pays $15.00/hr, and she opened a savings account with an initial deposit of $400 on January 15. She arranged for direct deposit of her payroll checks. If spring break begins March 20 and the trip will cost approximately $2,500, how many hours will she have to work to earn enough to pay for her vacation? If she can only work 4 hours per day, how many days per week will she have to work? How many weeks will it take? In this section, we will investigate problems like this and others, which generate graphs like the line in [link] .

Solving linear equations in one variable

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 $ax+b=0$ 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 $x$ 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 $5x+2=3x-6,$ we have the following:

The solution set consists of one number: $\left\{-4\right\}.$ 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 $5x-15=5\left(x-4\right),$ we have the following:

Indeed, $\mathrm{-15}\ne \mathrm{-20.}$ 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.