Graphing Lines
A linear equation has the general form Ax + By = C. The graph of this type of equation will look
like a straight line. There are different methods to graph this type of equation. This handout will
only present some of the most common.
1. Table of values– The most straight forward way to graph any equation is to make a table of
x values versus y values, once a table is generated, the points are plotted on a coordinate
system and a line is drawn through the points.
Example: Graph 3x – 4y = 12
Choose any values for x that are convenient. Then substitute (one at a time) these values into
the equation to find the corresponding y value.
The table will look similar to the
following:
x
-4
-1
− 154
0
4
Plotting these values on a coordinate
y
system yields
4
3
y
–6
or − 3 34
–3
0
2
1
0
-5
-4
-3
-2
-1
-1
0
1
2
3
4
5
-2
Note: Your table may differ if different
values for x were chosen.
-3
-4
-5
-6
2. x and y intercepts– A quick and easy way to graph a line is to identify the x and the y
intercepts. For any function, the y-intercept can be found by setting x = 0, and solving the
resulting equation. The converse is true for finding the x-intercept.
Following the directions above, gives a slimmed down version of the table above.
x–intercept
x
4
0
y
0
–3
y–intercept
These two points can be plotted and connected with a straight line to give the same graph shown
above. This method is quicker and simpler, but it is easier to make a mistake since the graph
relies only on two points.
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3. Slope-intercept method– This method requires that the equation first be converted from
general form (Ax + By = C) to standard form. Standard form for a linear equation is
y = mx + b. Once the equation is written in standard form, the coefficient of the x variable
gives the slope (m). The constant gives the y–intercept (b).
Δy ⎞
⎛
Slope ⎜ m =
⎟ – This equation indicates that the slope is a ratio of the change in y to the
Δx ⎠
⎝
change in x. In other words, the slope is a ratio of how much a line goes up or down (rise) to
rise(b) +−
how much a line goes left or right (run) or more commonly m =
. Therefore, if
run( − ↔ + )
the numerator of the slope is positive, the line goes up and if the numerator of the slope is
negative, the line will go down. Similarly, if the denominator of the line is positive the line
will go to the right and if the denominator is negative, the line will go to the left.
Example: Graph 3x – 4y = 12 using the slope and y-intercept.
First solve the equation for y.
Subtract 3x from both sides of the equation
3x – 4y = 12
–3x
–3x
– 4y = – 3x + 12
− 4 y − 3 x + 12
=
−4
−4
y=
3
x−3
4
Divide both sides of the equation by – 4
simplify both sides by reducing the fractions
the slope is ¾ and the y–intercept is – 3
y
Step 4
Connect the dots
with a straight line
Step 3
‘run’
right 4 units
x
Step 2
‘rise’
up 3 units
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