• linear equation s 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:
x =y
x2 − y = 3
3x − 4y = 6
x = 3
y = −2.
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.
d = 1720 − .50p
What are the p- and d-intercepts? What do they mean here?
The slope of a linear equation: the price-demand
equation
d = 1720 − .50p
If price increases by $1 how much does demand decrease?
Does this depend on the starting price?
If price increases by $1000 how much does demand decrease?
Does this depend on the starting price?
Is the decrease in demand always .50 times the increase in
price?
Slope of a line:
The slope or steepness of a straight line is the same between
any two points on the line. We can re formulate this property of
a line in algebraic terms as fol lows :
For any two points (x1, y1), (x2, y2) on the line, the ratio of the
change in y (change in y is y2 − y1) to the change in x (x2 − x1)
is the same no matter which two points on the line are chosen.
This common ratio is the slope of the line.
Definition 3 The slope of a line is defined as
where (x1, y1) and (x2, y2) are ANY two points on the line.
What is the slope of the line defined by the linear equation
4x − 3y = 12?
What is the slope of the line defined by the linear equation
d = 1720 − .50p?
Slope of the special cases:
vertical and horizontal lines
Slope formula: 
where (x1, y1) and (x2, y2) are ANY
two points on the line.
What is the slope of the line defined by the linear equation
y = −4?
What is the slope of the line defined by the linear equation
x = 3?
