6.2 Graphing linear inequalities

Overview

• Solutions and Lines
• General form of a Linear Equation
• The Intercept Method of Graphing
• Graphing Using any Two or More Points
• Slanted, Horizontal, and Vertical Lines

Solutions and lines

We know that solutions to linear equations in two variables can be expressed as ordered pairs. Hence, the solutions can be represented by point in the plane. We also know that the phrase “graph the equation” means to locate the solution to the given equation in the plane. Consider the equation $y-2x=-3$ . We’ll graph six solutions (ordered pairs) to this equation on the coordinates system below. We’ll find the solutions by choosing $x\text{-values}$ (from $-1\text{to}+4$ ), substituting them into the equation $y-2x=-3$ , and then solving to obtain the corresponding $y\text{-values}$ . We can keep track of the ordered pairs by using a table.

$y-2x=-3$

We have plotted only six solutions to the equation $y-2x=-3$ . There are, as we know, infinitely many solutions. By observing the six points we have plotted, we can speculate as to the location of all the other points. The six points we plotted seem to lie on a straight line. This would lead us to believe that all the other points (solutions) also lie on that same line. Indeed, this is true. In fact, this is precisely why first-degree equations are called linear equations.

Linear equations produce straight lines

General form of a linear equation

General form of a linear equation in two variables