Section 3.2: Graphing Linear Equations In Two Variables
§1 Graphing By Plotting Ordered Pairs
By now you should be comfortable plotting points. Now, we want to take things one step further and graph the
solutions to a linear equation. Let’s think about this for a bit. Previously, we saw that for any linear equation we
could multiple ordered pair solutions by making a table. Can we plot those ordered pair solutions on the x-y
plane? Of course we can! It turns out that every point on the line represents a solution to the linear equation,
hence every solution of the equation corresponds to a point on the line.
For example, find three solutions to the linear equation y  2x  1 . We can make table of values and see that
three possible solutions are (0,1),(2,5) and (3,7). If you plot these points and connect them, you should get a
straight line!
Remember, every point on the line represents a solution to the linear equation. Hence, we see that (-1,1) is also
a solution, and (-1/2,0) is a solution as well. For the most part, when graphing we only need two points.
PRACTICE
1) Graph the linear equation 3x  y  9
2) Graph the linear equation y  3x  2
§2 Intercepts
The word intercept really means to cross. Hence when graphing, an intercept represents the point on the graph
where the graph crosses the axes. Since there are two axes, there are two intercepts – the x-intercept and the yintercept. At the y-intercept, the x-coordinate is 0. Similarly for the x-intercept, the y-coordinate is zero. Hence
the intercepts should be easy to find. We use the following method:
PRACTICE
3) Find the intercepts of 5x  2 y  10 and graph.
4) Find the intercepts of y 
3
x  6 and graph.
2
§3 Graphing Linear Equations That Pass Through The Origin
Recall that the origin is simply the point (0,0). Hence if the line passes through the origin, then the ordered pair
(0,0) must be a solution to the equation. It turns out that equations of the form Ax  By  0 will always have
(0,0) as a solution, hence the graph will always pass through the origin! For example, if you try to graph
3x  y  0 , you will find that the ordered pair (0,0) is a solution! Can you find another solution? If you make a
table of values, you will see that (1,3) and (2,6) are two of the many solutions.
PRACTICE
5) Sketch the graph of 4 x  2 y  0
§4 Vertical And Horizontal Lines
These are special types of lines. If you look at the standard form of the equation of the line Ax  By  C , it turns
out that if A = 0, then the line will be a horizontal line. There will be no ‘x’ term in the equation. Hence the
equation can look something like 3 y  6 or simply y  2. What does this look like? It is the horizontal line that
has a y-intercept of 2!
Similarly, in the standard form of the equation, if B=0, then the line will be a vertical line. There will be no ‘y’
term in the equation. Hence the equation will look something like x  3 or something similar. This will be a
vertical line that has a x-intercept of -3.
PRACTICE
6) Sketch the graph of x = 5
7) Sketch the graph of y = -1