Call Now: (800) 537-1660
 Home    Why?     Free Solver   Testimonials    FAQs    Product    Contact

May 19th

May 19th

# Linear Equations and Graphing

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?

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!!

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 .

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?

 Prev Next

Home    Why Algebra Buster?    Guarantee    Testimonials    Ordering    FAQ    About Us
What's new?    Resources    Animated demo    Algebra lessons    Bibliography of     textbooks