Graphing Linear Equations

Graphing Linear Equations top left imageGraphing Linear Equations top right image
HomeAlgebraLinear Equations

Graphing Linear Equations is a mapping process to diagram straight lines as objects. Mathematicians often diagram using Cartesian Coordinate Systems.

A Cartesian Coordinate System provides a central point of reference for an object and between objects. Cartesian Coordinate Systems are drawn by a horizontal “x” line bisecting a “y’ line forming a 90 degree right angle. Coordinates “x” and “y” are on a single plane, a flat surface.

The slope of a straight line is a measure of steepness. Slope is described by a change in Rise, divided by a change in Run.

Slope = Rise / Run







“To graph or solve Linear Equations use the Slope-Intercept Form:

y = mx + b

Where m is the line slope, x is a coordinate value along the x axis and b is the y intercept.”

Many run and rise together look like stairs.

Rise is a distance Δy as y2 - y1, and Run is a distance Δx as x2 - x1 (Reference diagram at right).

Suppose coordinates (8, 7) and (4, 4), the slope, Δy / Δx, is:

(y2 - y1) / (x2 - x1) =

(7 - 4) / (8 - 4) =


Δy / Δx line slope

Graph of line showing linear slope as Δy / Δx.

How to Graph Linear Equations

By plugging values of “x” into a Linear Equation gives values of “y”.

Several coordinate solutions for y = 3x + 2:

“Y” values for “x” integer values -3 thru 3.

x and y cartesian coordinate solution set.

Choose two x and y coordinate solutions. Use them as points through which to plot the line.

Graph of coordinates for linear equation.

Graph of straight line equation y = 3x + 2 using coordinates (1, 5) and (-2, -4).

From the image directly above we already know that (1, 5) and (-2,  -4) are two coordinate points of Linear Equation y = 3x + 2. But, what if we only knew the two coordinate points, not the equation? Yes, we can create the equation by applying the Slope-Intercept Form y = mx + b:

First determine the line slope…

m = (y2 - y1) / (x2 - x1) = (5 - (-4)) / (1 - (-2)) = 9 / 3 = 3

and plug the slope, m , into the Slope-Intercept Equation y = mx + b.

y = 3x + b

Next determine b, the y intercept. To do this use any coordinate along the line and plug the x and y values into the Slope-Intercept Equation (I’m going to use coordinate (1, 5) )…

5 = 3 (1) + b

5 - 3 = b

2 = b

And we can now write y = 3x + 2, the Linear Equation defined by line segment coordinate points (1, 5) and (-2, -4).

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