How to Graph Linear Equations & Inequalities

By Yvonne Bernard
Cartesian plane
diagram 9 image by Yuriy Panyukov from

Graphing linear equations and inequalities is a useful tool in business as well as many disciplines of science. Graphing can represent data visually or predict trends or geometrically provide pictures of changes of one quantity with respect to another. Graphing with a coordinate system dates back to the second century B.C. and is credited to René Descartes. The Cartesian coordinate system or Cartesian plane is constructed using two real lines intersecting at right angles. The horizontal line is the x-axis and the y-axis is the vertical line. The point of intersection of these two axes is the origin.

How to Graph Linear Equations

Step 1

Draw a Cartesian coordinate plane on graph paper. Draw two intersecting perpendicular lines. Label the origin, the point of intersection, as (0,0).

Step 2

Look at your linear equation. The equation may be in standard form, Ax + By = C, or in some other common form. Get the equation into slope intercept form: y = mx+b. This may require manipulating the equation algebraically. For example if the equation reads: y - 3x = 2, algebraically manipulate the equation. First add 3x to both sides of the equation and it then becomes, y = 3x + 2. This is now in slope intercept form.

Step 3

Determine the \"y\" intercept, the number that corresponds to the (b) in the slope intercept form. This is where the graph crosses the y axis. The x coordinate of this point is zero. In the example above, y = 3x + 2, the y intercept is 2.

Step 4

Determine the slope (m). In the example above the term that corresponds to the slope or m is 3. This is the term in front of the x. Slope is defined as rise over run, or the ratio of y to x.

Step 5

Plot the \"'y\" intercept on the Cartesian plane. Take a pencil and physically make a dot on the plane corresponding to the determined \"'y\"intercept. In the example since the \"'y\" intercept is 2, this would be two ticks above the zero directly on the y-axis.

Step 6

Plot the second point of the line using the slope. Place the pencil on the first point. In the example the slope is 3 or more exactly, 3/1, therefore move the pencil as you count three ticks up (rise or \"y\") and then move the pencil over one tick to the right (run or \"x\"). At this spot make another dot.

Step 7

Draw a line through the two points with a straightedge. This is the graph of the given linear equation .

Graphing Inequalities

Step 1

Replace the inequality symbol in your equation with an equal sign. For example if given y > 3x + 2, rewrite as y = 3x + 2 and repeat steps 1 through 6 in the first section.

Step 2

Draw a dashed line through both points if the initial inequality contains a greater than (>)or a less than (<) symbol.

Step 3

Draw a solid line through both points if the inequality contains a greater than or equal (?), or a less than or equal (?) sign.

Step 4

Choose a test point to determine which side of the line should be shaded. If the point chosen makes the inequality true then the area that contains that point should be shaded and vice versa. In the example if the point (2,1) were chosen you would plug in the values of two and one for x and y respectively in the original inequality and see if this point makes the inequality true. In this case you would have, 1 > 3 (2) + 2 or 1 > 6 + 2 or 1 > 8. This is not a true statement, one is not greater than eight. Thus that point and all points on the same side of the line as this test point will not be shaded or included in the solution set.

About the Author

Yvonne Bernard began her writing career in 1994. She has written articles for “Design News” and published an engineering text book, “Environmental Engineering,” for McGraw-Hill. Bernard is also a member of several engineering societies. She has a Bachelor of Science in mechanical engineering from the University of Illinois.