Chapter 1 Review Notes

Transcription

Chapter 1
Linear Equations and Straight
Lines
Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/Siegel
Copyright © 2010 Pearson Education, Inc.
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Outline
1.1 Coordinate Systems and Graphs
1.2 Linear Inequalities
1.3 The Intersection Point of a Pair of Lines
1.4 The Slope of a Straight Line
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Coordinate Plane
Construct a Cartesian
coordinate system on a
plane by drawing two
coordinate lines, called
the coordinate axes,
perpendicular at the
origin. The horizontal
line is called the x-axis,
and the vertical line is
the y-axis.
y
y-axis
O
Origin
x
x-axis
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Coordinate Plane: Points
Each point of the plane is identified by a pair of
numbers (a,b). The first number tells the number
of units from the point to the y-axis. The second
tells the number of units from the point to the xaxis.
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Example Coordinate Plane
Plot the points: (2,1), (-1,3), (-2,-1) and (0,-3).
(-1,3)
-1 y
3
2
(2,1)
1
x
-1
(-1,-2)
-2
-3
(0,-3)
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Example Graph of an Equation
Sketch the graph of the equation y = 2x - 1.
y
(2,3)
x
y
-2
2(-2) - 1 = -5
-1
2(-1) - 1 = -3
0
2(0) - 1 = -1
1
2(1) - 1 = 1
2
2(2) - 1 = 3
(1,1)
x
(0,-1)
(-1,-3)
(-2,-5)
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Linear Equation
An equation that can be put in the form
cx + dy = e
(c, d, e constants)
is called a linear equation in x and y.
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Standard Form of Linear Equation
The standard form of a linear equation is
y = mx + b
(m, b constants)
if y can be solved for, or
x=a
(a constant)
if y does not appear in the equation.
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Example Standard Form
Find the standard form of 8x - 4y = 4 and 2x = 6.
(a) 8x - 4y = 4
(b) 2x = 6
8x - 4y = 4
- 4y = - 8x + 4
y = 2x - 1
2x = 6
x=3
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Intercepts
x-intercept: a point on the graph that has a ycoordinate of 0. This corresponds to a point
where the graph intersects the x-axis.
y-intercept: the point on the graph that has a xcoordinate of 0. This corresponds to the point
where the graph intersects the y-axis.
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Graph of y = mx + b
To graph the equation y = mx + b:
1. Plot the y-intercept (0,b).
2. Plot some other point. [The most convenient
choice is often the x-intercept.]
3. Draw a line through the two points.
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Example Graph of Linear Equation
Use the intercepts to graph y = 2x - 1.
x-intercept: Let y = 0
y
0 = 2x - 1
(1/2,0)
x = 1/2
x
y-intercept: Let x = 0
y = 2(0) - 1 = -1
(0,-1)
y = 2x - 1
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1.2 Linear Inequalities
1.
2.
3.
4.
5.
6.
7.
Definitions of Inequality Signs
Inequality Property 1
Inequality Property 2
Standard Form of Inequality
Graph of x > a or x < a
Graph of y > mx + b or y < mx + b
Graphing System of Linear Inequalities
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Graph of x > a or x < a
The graph of the inequality
 x > a consists of all points to the right of and
on the vertical line x = a;
 x < a consists of all points to the left of and on
the vertical line x = a.
We will display the graph by crossing out the
portion of the plane not a part of the solution.
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Example Graph of x > a
Graph the solution to 4x > -12.
First write the equation in standard form.
y
4x > -12
x = -3
x > -3
x
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Graph of y > mx + b or y < mx + b
To graph the inequality, y > mx + b or
y < mx + b:
1. Draw the graph of y = mx + b.
2. Throw away, that is, “cross out,” the portion of
the plane not satisfying the inequality. The graph
of y > mx + b consists of all points above or on
the line. The graph of y < mx + b consists of all
points below or on the line.
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Example Graph of y > mx + b
Graph the inequality 4x - 2y > 12.
First write the equation in standard form.
4x - 2y > 12
y
- 2y > - 4x + 12
y < 2x - 6
x
y = 2x - 6
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Example Graph of System of Inequalities
Graph the system of inequalities
2 x  3 y  15

4 x  2 y  12

y  0.

The system in standard form is

y 

y 



2
x  5
3
2x  6
y
 0.
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1.3 The Intersection Point of a Pair of Lines
1.
2.
3.
4.
Solve y = mx + b and y = nx + c
Solve y = mx + b and x = a
Supply Curve
Demand Curve
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Solve y = mx + b and y = nx + c
To determine the coordinates of the point of
intersection of two lines
y = mx + b and y = nx + c
1. Set y = mx + b = nx + c and solve for x. This
is the x-coordinate of the point.
2. Substitute the value obtained for x into either
equation and solve for y. This is the y-coordinate
of the point.
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Example Solve y = mx + b & y = nx + c
Solve the system
2 x  3 y  7

4 x  2 y  9.
Write the system in standard form, set equal
and solve.
2
7

 y  3 x  3

 y  2x  9

2
2
7
9
y
x   2x 
3
3
2
8
41
x
3
6
41
x
16
 41  9 5
y  2   
 16  2 8
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Example Point of Intersection Graph
Point of Intersection: (41/16, 5/8)
y
y = 2x - 9/2
(41/16,5/8)
x
y = (-2/3)x + 7/3
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1.4 The Slope of a Straight Line
1.
2.
3.
4.
5.
6.
Slope of y = mx + b
Geometric Definition of Slope
Steepness Property
Point-Slope Formula
Perpendicular Property
Parallel Property
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Slope of y = mx + b
For the line given by the equation
y = mx + b,
the number m is called the slope of the line.
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Example Slope of y = mx + b
Find the slope.
y = 6x - 9
m=6
y = -x + 4
m = -1
y=2
m=0
y=x
m=1
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Geometric Definition of Slope
Geometric Definition of Slope Let L be a line
passing through the points (x1,y1) and (x2,y2)
where x1 ≠ x2. Then the slope of L is given by the
formula
y2  y1
m
.
x2  x1
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Example Geometric Definition of Slope
Use the geometric definition of slope to find the
slope of y = 6x - 9.
Let x = 0. Then y = 6(0) - 9 = -9.
(x1,y1) = (0,-9)
Let x = 2. Then y = 6(2) - 9 = 3.
(x2,y2) = (2,3)
3   9 
12
m
 6
20
2
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Steepness Property
Steepness Property Let the line L have slope
m. If we start at any point on the line and move 1
unit to the right, then we must move m units
vertically in order to return to the line. (Of
course, if m is positive, then we move up; and if
m is negative, we move down.)
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Example Steepness Property
Use the steepness property to graph
y = -4x + 3.
The slope is m = -4.
A point on the line is (0,3).
If you move to the right 1
unit to x = 1, y must move
down 4 units to y = 3 - 4 = -1.
y
(0,3)
x
(1,-1)
y = -4x + 3
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Point-Slope Formula
Point-Slope Formula
The equation of the
straight line through the point (x1,y1) and having
slope m is given by
y - y1 = m(x - x1).
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Example Point-Slope Formula
Find the equation of the line through the point
3
(-1,4) with a slope of  .
5
Use the point-slope formula.
3
y  4    x   1 
5
3
3
y4  x
5
5
3
17
y   x
5
5
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Perpendicular Property
Perpendicular Property When two lines are
perpendicular, their slopes are negative
reciprocals of one another. That is, if two lines
with slopes m and n are perpendicular to one
another, then
m = -1/n.
Conversely, if two lines have slopes that are
negative reciprocals of one another, they are
perpendicular.
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Example Perpendicular Property
(3,-5) that is perpendicular to the line whose
equation is 2x + 4y = 7.
The slope of the given line is -1/2.
The slope of the desired line is -(-2/1) = 2.
Therefore, y -(-5) = 2(x - 3) or
y = 2x – 11.
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Parallel Property
Parallel Property
Parallel lines have the
same slope. Conversely, if two lines have the
same slope, they are parallel.
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Example Parallel Property
(3,-5) that is parallel to the line whose
equation is 2x + 4y = 7.
The slope of the given line is -1/2.
The slope of the desired line is -1/2.
Therefore, y -(-5) = (-1/2)(x - 3) or
y = (-1/2)x - 7/2.
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Graph of Perpendicular & Parallel Lines
2x + 4y = 7
y = 2x - 11
y = (-1/2)x - 7/2
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Chapter 1 Review Notes

Transcription

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