Linear Equations and Graphing

• Linear Equations in two variables

• Graphing Ax+By = C

• Slope of a line

• Special Forms of a linear equation

• More applications

The Price-demand equation again: d = 1720 − .50p. We can
represent many solutions on a graph.

Definition 1 A linear equation in two variables is an equation
that can be written in the standard form

Ax+By = C

where A, B, and C are constants (A and B not both zero) and
x and y are variables.

Circle the LINEAR equations:

Do you think that all such linear equations, when graphed will
give a straight line? Why? Let’s try some examples:

What about x − y = 0?
What about x = 3?
What about y = −1?

The shape of the graph of a linear equation:

Theorem 1 The graph of any equation of the form Ax+By = C
is a line, and any line in the cartesian coordinate system is the
graph of an equation of this form.

This theorem makes it very easy to graph a linear equation...
For example if we know two points on the graph we are done!!

Moral: Theorems are your friends.

Two special cases:

• Horizontal lines y = c for some number c

• Vertical lines x = c for some number c

Try graphing 4x − 3y = 12 by finding two easy solutions to the
equation and plotting those.

What about x = 3?
What about y = −4?

x− and y− intercepts of an equation:

Definition 2 In the graph of any equation of two variables, the
points where the graph of an equation crosses the x-axis are
called the x-intercepts and the points where the graph crosses
the y-axis are called the y-intercepts.

What do the coordinates of an x-intercept look like?
To find them set   = 0 and solve for .

What do the coordinates of a y-intercept look like?
To find them set    = 0 and solve for .

The intercepts of a linear equation are easier to locate than the
intercepts of most other equations.

x- and y-intercepts of a linear equation:

Find the x- and y- intercepts for 4x − 3y = 12.

Find the x- and y-intercepts for 7x − .2y = 12.

Could a linear equation have more than one x-intercept?

Does a linear equation always have an x-intercept? CAREFULL
HERE!

Intercepts for the Price-demand equation.

What are the p- and d-intercepts? What do they mean here?