Graph of Linear Equations

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Graph of Linear Equations

• Objective
• Graph linear equations

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The graph of any linear equation is a straight
line.
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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.

• The y-intercept of a graph is the
• y-coordinate of the point where the graph
  intersects the y-axis.

(0,4)
(5,0)
The x-intercept is the x-coordinate of the points
where the graph crosses the x-axis.
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Graph 4x 5y 20

• First find the intercepts.
• To find the y-intercept, let x 0 and solve for
  y. we find ____?

y4
We plot the point (0,4).
To find the x-intercept, let y0 and solve for x.
we find _____?
x5
We plot the point (5,0).
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Theorem 2 The graph of ymx is a line
containing the origin. The graph of ymxb is a
line parallel to ymx and has b as the
y-intercept.
Y2x
Y2x - b
(0,b)
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AssignmentGraph and compare the graph of y2x
with 1. Y2x1 and y2x-4
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Section 2Objective 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.

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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)
Y4
X
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Exercise Graph these equations.

• X 4
• Y -3
• Y 0

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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.

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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.

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How will / can we find the slope of a line
containing a given pair of points???
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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
Where (x1, y1) (x2,y2) are any two points on
the line x2 is not equal to x1
We usually use the letter m to designate slope.
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Example

• The points (1,2) (3,6) are on a line, find its
  slope

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What if we use the opposite order given example?
Do we get the same slope??

• Illustrate show your solution.

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• 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

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The points (0,0) (-1,-2) are also on the line if
we compute for the slope , we get the same
answer, correct??
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.
(3,6)
(1,2)
(-1,-2)
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m is positive number greater than 1
m5
m is positive between 1 and 0
m1
m1/4
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Section 4Objective Find the slope of a
horizontal vertical line.
Find the slope of the line y 3?
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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.
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Exercise Find the slope, if it exists.

• Y -5
• X 17

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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.
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Theorem 5.The Point Slope Equation

• A line containing (x1,y1) with slope m has an
  equation (y - y1)m (x - x1)

Example Find an equation of a line containing
(1/2 , -1) with slope 5.
(y - y1)m (x - x1) (y (-1) 5(x ½) y1 5(x
½) y5x-7/2
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Example Find an equation of the line with
y-intercept 4, with slope 3/5.

• (y - y1)m (x - x1)
• y-43/5(x-0)
• y3/5x 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.
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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 pointslope equation, we obtain the two-point
  equation.

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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

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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).
• Soln y-3 -4 3 (x-2)
• 1 2
• y 7x 11

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Question if we use the other pair, do we still
get the same equation??

• You try on your notebook!!!

YES!!!
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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)

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Section 7

• Objective Find the slope and y-intercept 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.
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Example Find the slope and y-intercept of the
line whose equation is y 2x-3

• y 2x-3
• Slope 2
• y-intercept 3

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Exercise

• 1.Find an equation of the line containing the
  point (21,9) with slope 1/32. 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 1/4xy
• 7. Find the slope and y-intercept of the line
  whose equation is -6y-3x7

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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??
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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
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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?
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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

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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

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Exercise

• Find the distance between the points
• (-5,3) and (2, -7)
• (3, 3) and (-3, -3)

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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.
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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

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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)
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SEATWORK!!!

• IN 1 WHOLE PAD PAPER
• ANSWER THE FOLLOWING!

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• 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 1/4xy
• 7. Find the slope and y-intercept of the line
  whose equation is -6y-3x7
• 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)