3.2 Graphing Linear Equations in Two Variables

3.2 Graphing Linear Equations in Two Variables
1. Recall:
a. A linear equation in two variables is an equation that can be written in
the form
Ax + By = C
where A, B, and C are real numbers and A and B not both zero. Note that
x and y are variables.
b. An ordered pair of real numbers is a pair of real numbers in which order
matters. Thus, there is a first number and a second number. (For
example: (2, –5.5) )
For equations in the two the variables, x and y, the x-value is always the
first number and the y-value is always the second number.
c. To graph an ordered pair (a, b) on a rectangular (or Cartesian)
coordinate system:
Locate the first number, a, on the x-axis (horizontal axis) and visualize a
vertical line through this number.
Move vertically up or down the number of units given by the second
number, b.
This (single) point is the point associated with the ordered pair (a, b).
Note that both numbers are needed to locate a single point in the plane.
These numbers are called the coordinates of the point.
d. A solution to a linear equation in the two variables, x and y, is an
ordered pair of real numbers (a, b) with the property that when a is
plugged into the equation for x and b is plugged into the equation for y,
a true statement is obtained.
e. There are infinitely many solutions to a linear equation in two variables.
f. To find a solution, plug any number in for one variable and solve the
resulting linear equation for the other variable.
2. The graph of a linear equation in two variables is the straight line formed by
plotting all ordered pair solutions to the linear equation. (Every point on this line
is a solution and every solution is a point on the line.)

3. Method: Because two distinct points determine a line, to graph a linear equation,
plot any two distinct ordered pair solutions and connect them with a line. (Plot a
third solution as a check.)
4. Don't pick points to graph that are too close together. (It would be hard to graph
an accurate line.)
5. Intercepts:
a. The x-intercept is the point where the graph crosses the x-axis.
Find the x-intercept by plugging zero on for the y coordinate and solving
the resulting equation for x. (Let y = 0.)
b. The y-intercept is the point where the graph crosses the y-axis.
Find the y-intercept by plugging zero on for the x coordinate and solving
the resulting equation for y. (Let x = 0.)
6. Lines through the Origin: If A and B are non-zero real numbers, the graph of
a linear equation of the form
Ax + By = 0
passes through the origin (0, 0). (Thus, to graph this type of linear equation, plot
the point (0, 0), which is the x- and y- intercept, and two other points. The x- and
y-intercepts will not be enough to graph these lines.)
7. Horizontal Line: A linear equation of the form y = b (or By + C = 0 ) has a
graph given by a horizontal line through the point (0, b).
8. Vertical Line: A linear equation of the form x = a (or 1⋅ x + 0⋅ y = a) has a
graph given by a vertical line through the point (a, 0).
9. See the summary of linear equations on page 212.
10. You can graph linear equations of the form Ax + By = C in a calculator by
first solving for y. Enter the resulting expression into one of the slots in the y=
key. Hit the GRAPH key. Check that the window shows the x- and y-intercepts.
If these intercepts do not show in the window, go to the WINDOW key and
change the x and y ranges. (The Standard window - #6 ZStandard in the ZOOM
key is often a place to start.)