Download Graph of Linear Equations

Graph of Linear
Equations

Objective:
– Graph linear
equations
The graph of any linear
equation is a straight
line.
Since two points determine a line, we can graph a linear
equation by finding two points that belong to the graph.
Then we draw a line containing those point. A third point
should always be used as a check. Often the easiest
points to find are the points where the graph crosses the
axes.

(0,4)
(5,0)
The y-intercept of a
graph is the
y-coordinate of the
point where the
graph intersects the
y-axis.
The x-intercept is the xcoordinate of the points where
the graph crosses the x-axis.
Graph 4x + 5y = 20

First find the
intercepts.
– To find the y-intercept,
let x = 0 and solve for y.
y=4
we find ____?
We plot the point
(0,4).
To find the x-intercept,
let y=0 and solve for x.
x=5
we find _____?
We plot the point
(5,0).
Theorem #2: The graph of
y=mx is a line containing the
origin. The graph of y=mx+b is
a line parallel to y=mx and has
b as the y-intercept.
Y=2x
Y=2x - b
(0,b)
Assignment:
Graph and compare the graph of
y=2x with 1. Y=2x+1 and y=2x-4
Section #2:
Objective: Graph the equation whose
graphs are parallel to the x-axis or y-axis

Theorem #3. For constant a and b of
an equation of the form y = b, is a line
parallel to the x-axis with y intercept b.
The graph of an equation of the form x
= a is a line parallel to the y-axis with
x-intercept a.
Example:
Any ordered pair (x,4) is a
solution thus the line is parallel
to the x-axis and the y-intercept
is 4.
Y
(0,4)
Y=4
X
Exercise: Graph these
equations.
1.X = 4
2.Y = -3
3.Y = 0
Graphing Linear
Equations:




1. If there is a variable missing, solve for the other
variable. The graph will be a line parallel to an axis.
2. If no variable is missing, find the intercepts. Use
the intercepts to graph.
3. If the intercept points are too close together, or
are the same point, choose another point farther the
origin.
4. Use a third point as a check.
Section III.

Objective:
– To find the slope of a line given two points
on it.
– To find the slopes of horizontal and
vertical lines.
– To find the point-slope form of the
equation of a line.
How will / can we
find the slope of a
line containing a
given pair of
points???
When we say slope!?
What comes to your
mind??

The ratio of the change in y to the
change in x - slope of a line.
DEFINITION: The slope m of a line is the change in
y divided by the change in x or
m = y2 – y1 / / x2- x1
We usually use the letter m
to designate slope.
Where (x1, y1) & (x2,y2) are
any two points on the line &
x2 is not equal to x1
Example:

The points (1,2) & (3,6) are on a line,
find its slope
What if we use the
opposite order given
example? Do we get
the same slope??

Illustrate show your solution.

When we compute the slope, the order
of the points DOES NOT matter as
long as we take the same order of
finding the difference
The points (0,0) (-1,-2) are also
on the line if we compute for
the slope , we get the same
answer, correct??
(3,6)
(1,2)
(-1,-2)
REMEMBER: if a line
slants up from the left to
right, it has a positive slope
if a line slants down
from left to right it has a
negative slope.
m is positive number greater
than 1
m=5
m is positive between 1
and 0
m=1
m=1/4
Section #4:
Objective: Find the
slope of a horizontal &
vertical line.
Find the slope of the line y = 3?
Vertical & Horizontal lines do not slant.
Any two points on a horizontal line have
the same y-coordinate. The change in y is
0, so the slope is 0.
Try on your notebook.
Find the slope of the line x = -4.
Theorem #4
A horizontal line has slope 0.
A vertical line has no slope.
Exercise: Find the slope, if it exists.
a. Y = -5
b. X = 17
Section #5

Objective:
– Use the point-slope equation to find an
equation of a line.
If we know the slope of a line and the coordinates of a
point on the line, we can find an equation of the line
point – slope equation.
Theorem #5.
The Point Slope Equation

A line containing (x1,y1) with slope m has
an equation (y - y1)=m (x - x1)
1(/ 2
Example: Find an equation of a (line
containing (1/2 , -1)
with slope 5.
(y - y1)=m (x - x1)
(y –(-1)= 5(x – ½)
y+1= 5(x – ½)
y=5x-7/2
Example: Find an equation of the line with y-intercept 4,
with slope 3/5.
(y - y1)=m (x - x1)
y-4=3/5(x-0)
y=3/5x + 4
Exercise:
1.Find an equation of the line containing the point
(-2,4) with slope –3
2. Find an equation of the line containing the point
(-4,-10) with slope ¼.
3. Find an equation of the line with x- intercept 5 and
slope –1/2.
Section #6
Objective:Given two points, we can find
an equation of the line containing
them. If we find the slope of a line
dividing the change in y by change in
x, and substitute this value for m in the
point=slope equation, we obtain the
two-point equation.
Theorem #6

The Two-Point Equation
– Any non-vertical line tine containing the
points (x1,y1) and (x2,y2) has an equation
y - y1 = y2-y1 (x-x1)
x2-x1
Example: Find an equation of the line
containing the points (2,3)and (1,-4),
We find the slope and then substitute in
the two-point equation. We take (2,3)
as (x1,y1) and (1,-4) as (x2,y2).
Sol’n:
y-3 = -4 –3 (x-2)
1 –2
y = 7x –11
Question: if we use the
other pair, do we still get
the same equation??
You try on your notebook!!!
YES!!!
Exercise:

Find an equation of the line containing
the following pairs of points.
– 1. (1,4) and (3,-2)
– 2. (3,-6) and (0,4)
– 3. (2,-5) and (7,1)
Section #7

Objective: Find the slope and yintercept of a line, given the slope
intercept equation for the line.
Theorem #7
The Slope-Intercept Equation
A non-vertical line with slope m
and y-intercept b has an equation y= mx+ b.
Example: Find the slope and yintercept of the line whose
equation is y = 2x-3
y = 2x-3
Slope 2
y-intercept –3
Exercise:
1.Find an equation of the line containing the point
(21,9) with slope –1/3
2. Find an equation of the line containing the point
(4,20) with slope ¼.
3. Find an equation of the line with x- intercept 3 and
slope 2.
4. Find an equation of the line containing the points
(3,3)and (9,-4),
5.Find an equation of the line containing the
following pairs of points.
1. (5,4) and (3,-8)
2. (8,-6) and (0,5)
3. (7,-5) and (7,3)
6. Find the slope and y-intercept of the line
whose equation is 9 = 1/4x+y
7. Find the slope and y-intercept of the line
whose equation is -6y=-3x+7
Section #8
Graphing Using Slope-Intercept Form
 Objective: Graph linear equation in
slope-intercept form

Example: Graph 5y – 20 = -3x
So HOW are we going to solve for
this??
Solution

Solve for y, we find the slope-intercept
form
y = -3/5 x + 4.
Thus, the y-intercept is 4 and the slope is –3/5
DISTANCE FORMULA

The distance formula can be used to
find the distance between two points
when we know the coordinates of the
points
ANY IDEA WHAT IS THE DISTANCE
FORMULA? REFRESH YOUR SECOND YEAR
TOPIC?
Theorem #10

The Distance Formula
The distance between any two points
(x1,y1) and (x2,y2) is given by
D = sqrt [(x1-x2)2 + (y1-y2) 2]
Example:
Find the distance between points (8,7)
and (3,-5)
D = sqrt [(x1-x2)2 + (y1-y2) 2]
D = sqrt [(8-3)2 + (7-(-5)) 2]
D = sqrt (25 + 144 )
D = sqrt (169)
D = 13

Exercise:
Find the distance between the points
1. (-5,3) and (2, -7)
2. (3, 3) and (-3, -3)
MIDPOINT OF
SEGMENTS
Objective: Find the coordinates of the
midpoint of a segment, given the
coordinates of the endpoints.
The coordinates of the midpoint of a segments can
be found by averaging the coordinates of the
endpoints. We can use the distance formula to verify
a formula for finding the coordinates of the
midpoint of a segment when the coordinates of the
endpoints are known.
Theorem #11

The Midpoint Formula
– If the coordinates of the endpoints of a
segment are (x1,y1) and (x2,y2), then the
coordinates of the midpoint are
[(x1+ x2) / 2 , (y1+ y2) / 2 ]
Example:

Find the coordinates of the midpoint of
the segment with endpoints (-3, 5) and
(4, -7)
Using the midpoint formula, we get
[(-3+ 4) / 2 , (5+(-7)) / 2 ]
(1/2 , -1)
SEATWORK!!!
IN 1 WHOLE PAD PAPER
ANSWER THE FOLLOWING!
1.Find an equation of the line containing the point (21,9) with
slope –1/3
2. Find an equation of the line containing the point
(4,20) with
slope ¼.
3. Find an equation of the line with x- intercept 3 and slope 2.
4. Find an equation of the line containing the points (3,3)and (9,4).
5.Find an equation of the line containing the following pairs of
points.a. (5,4) and (3,-8)
b. (8,-6) and (0,5)

c. (7,-5) and (7,3)
6. Find the slope and y-intercept of the line whose equation is 9
= 1/4x+y

7. Find the slope and y-intercept of the line whose equation
is -6y=-3x+7
8. Find the distance between the points
a. (-7,3) and (2, -7)
b. (2, 2) and (-2, -2)
9. Find the coordinates of the midpoint of the segment with
endpoints (3, -5) and (4, -7)