GRAPHING LINEAR EQUATIONS
A linear equation can be defined as an equation in which the highest exponent of the equation
variable is one. When graphed, the equation is shown as a single line. A linear equation has only
one solution. The solution of a linear equation is equal to the value of the unknown variable that
makes the linear equation true.
A linear equation in the form of ax + by = c is said to be in standard form. A linear equation in the
form of =
y mx + b is said to be in slope intercept form. The solution set for linear equations in
either form is not just a single number but a set of ordered pairs of numbers ( x, y ) . The solution set
can be represented graphically on a Cartesian coordinate system, and its graph is a straight line.
Graphing a Linear Equation in Standard Form
The easiest way to graph a linear equation in standard form is by plotting its points. Different set of
points can be found by giving one of the variables a value and solving for the other variable. At the
end all the points can be plotted in the graph and joined together by a straight line.
Example:
Draw a graph of 3 x + 2 y =
12
Step 1 – Assign values to one unknown and calculate the corresponding
values
for the other unknown.
x
0
2
4
y
6
3
0
Step 2 – Plot these points on a graph. Connect the points with a straight line.
If the points do not lie in a straight line there is probably an error in
one of the points used.
y
7
6
5
4
3
2
1
x
−3
−2
−1
−1
−2
−3
4
1
2
3
4
5
6
Graphing Linear Equations in Slope Intercept Form
Linear equations in slope intercept form can also be graphed using the points system. However, the
form in which the equation is written clearly tells us the slope and the y-intercept and thus makes it
easier to plot with just two points. Remember that in the form y = mx + b , m is the slope and b is the
y-intercept. This is the slope intercept form.
Example:
Graph 3 x + 2 y = 12
Step 1 – Since the equation is in standard form, solve for y to get the equation
in
slope-intercept form =
y mx + b
y=
3
− 3 x + 12 − 3 x 12
=
− x+6
=
+
2
2
2
2
Step 2 – Remember that in the slope-intercept form equation,=
y mx + b , m is
the slope and b is the y-intercept. By looking at the equation we can
know our slope and y-intercept:
m= −
3
2
b=6
→ Slope
→ y-intercept
Step 3 – Using the slope and the y-intercept, graph the equation of the line.
Start
by finding the y-intercept on the graph. From there find the slope.
Remember that the slope is rise over run, in this case the slope is
m = − 3 2 , which means that from the y-intercept point we need to
move three spaces down since the slope is negative and then two
places to the right. Remember that you always move to the right.
The negative sign does not indicate moving to the left; it is only an
indicator for going up or down. Always move to the right.
y
9
8
7
slope – starting from your y-intercept, move three
6
y-intercept
(0, 6)
units down (because of negative sign), and then two
units to the right m = − 3 2
5
4
3
2
1
x
−4 −3 −2 −1
−1
−2
−3
−4
5
1
2
3
4
5
6
7
8
9 10
Step 4 – After you have found the two points, draw a straight line connecting
the
two points together. This is the graph of your equation.
y
9
8
7
draw a connecting line between the two
points obtained
6
5
4
3
2
1
x
−4 −3 −2 −1
−1
−2
−3
−4
5
1
2
3
4
5
6
7
8
9 10