Chapter 1 linear equations and straight lines

Chapter 1

Linear Equations and
Straight Lines

Copyright © 2014, 2010, 2007 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
1.5 The Method of Least
Squares


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Section 1.1

Coordinate Systems
and Graphs


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Outline
1.
2.
3.
4.
5.

Coordinate Line
Coordinate Plane
Graph of an Equation
Linear Equation
Standard Form of Linear
Equation
6. Graph of x = a
7. Intercepts
8. Graph of y = mx + b

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Coordinate Line
Construct a Cartesian coordinate system on a
line by choosing an arbitrary point, O (the
origin), on the line and a unit of distance along
the line. Then assign to each point on the line a
number that reflects its directed distance from
the origin.


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Example Coordinate Line
Graph the points -3/5, 1/2 and 15/8 on a
coordinate line.
1/2

-3/5

-4

-3

-2

-1
Origin

0

15/8

1

2

3

4

Unit
length
Positive numbers

Negative numbers


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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
x-axis

Origin


x

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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
x-axis.


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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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Graph of an Equation
The collection of points (x,y) that satisfies an
equation is called the graph of that equation.
Every point on the graph will satisfy the equation
if the first coordinate is substituted for every
occurrence of x and the second coordinate is
substituted for every occurrence of y in the
equation.


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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(0) - 1 = -1

1

2(1) - 1 = 1

2

2(2) - 1 = 3

x

2(-1) - 1 = -3

0

(1,1)

(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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Graph of x = a
The equation x = a graphs into a vertical line a
units from the y-axis.
y

y
x=2

x = -3

x


x

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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
x = 1/2

(1/2,0)
x

y-intercept: Let x = 0
y = 2(0) - 1 = -1

(0,-1)
y = 2x - 1


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Summary Section 1.1
 Cartesian coordinate systems associate a
number with each point of a line and associate a
pair of numbers with each point of a plane.
 The collection of points in the plane that
satisfy the equation ax + by = c lies on a straight
line. After this equation is put into one of the
standard forms y = mx + b or x = a, the graph is
easily drawn.

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Section 1.2

Linear Inequalities


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Outline
1. Definitions of Inequality
Signs
2. Inequality Property 1
3. Inequality Property 2
4. Standard Form of
Inequality
5. Graph of x > a or x < a
6. Graph of y > mx + b or y <
mx + b
7. Graphing System of
Linear Inequalities

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Definitions of Inequality Signs
 a < b means a lies to the left of b on the
number line.
 a < b means a = b or a < b.
 a > b means a lies to the right of b on the
number line.
 a > b means a = b or a > b.


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Inequality Signs Example
-4

-3

-2

-1

0

1

2

3

4

Which of the following statements are true?
True
1<4
True
-1 > -4
True
2<3
True
0 > -2
True
3>3

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Inequality Property 1
Suppose that a < b
and that c is any number. Then a + c < b + c. In
other words, the same number can be added or
subtracted from both sides of the inequality.

Note: Inequality Property 1 also holds if < is
replaced by >, < or >.


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Example Inequality Property 1
Solve the inequality x + 5 < 2.
Subtract 5 from both sides to isolate the x on the
left.
x+5<2
x+5-5<2-5
x < -3
The values of x for which the inequality holds
are exactly those x less than or equal to −3.

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2A. If a < b and c is positive, then ac < bc.
2B. If a < b and c is negative, then ac > bc.

Note: Inequality Property 2 also holds if < is
replaced by >, < or >.


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Example Inequality Property 2
Solve the inequality -3x + 1 > 7.
Subtract 1 from both sides to isolate the x term on the
left.
-3x + 1 > 7
-3x + 1 - 1 > 7 - 1
-3x > 6
Divide by -3, or multiply by -1/3 to isolate the x.
x < -2


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Standard Form of Linear Inequality
A linear inequality of the form cx + dy < e
can be written in the standard form
1. y < mx + b or y > mx + b if d ≠ 0, or
2. x < a or x > a if d = 0.
Note: The inequality signs can be replaced by
>, < or >.


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Example Linear Inequality Standard Form
Find the standard form of 5x - 3y < 6 and 4x > -8.
(a) 5x - 3y < 6

(b) 4x > -8

5x - 3y < 6
-3y < - 5x + 6
y > (5/3)x - 2

4x > -8
x > -2


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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
2x

15

4x

2y

12

y

Graph the system of inequalities

3y

0.

The system in standard form is
y
y

2
x
3
2x
y

5
6
0.


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Summary Section 1.2 - Part 1
 The direction of the inequality sign in an
inequality is unchanged when a number is added
to or subtracted from both sides of the inequality,
or when both sides of the inequality are
multiplied by the same positive number. The
direction of the inequality sign is reversed when
both sides of the inequality are multiplied by the
same negative number.


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 The collection of points in the plane that
satisfy the linear inequality ax + by < c or
ax + by > c consists of all points on and to one
side of the graph of the corresponding linear
equation. After this inequality is put into
standard form, the graph can be easily pictured
by crossing out the half-plane consisting of the
points that do not satisfy the inequality.

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The feasible set of a system of linear inequalities (that
is, the collection of points that satisfy all the
inequalities) is best obtained by crossing out the points
not satisfied by each inequality. The feasible set
associated to the system of the previous example is a
three-sided unbounded region.


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Section 1.3

The Intersection Point
of a Pair of Lines


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Outline
1. Solve y = mx + b and y = nx
+c
2. Solve y = mx + b and x = a
3. Supply Curve
4. 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

2x

3y

7

4x

2y

9.

Write the system in standard form, set equal
and solve.
y
y
y

2
x
3
2x
2
7
x
3
3

7
3
9
2
9
2x
2

8
41
x
3
6
41
x
16
41 9
y 2
16
2


Slide 41

5
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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Solve y = mx + b and x = a
To determine the coordinates of the point of
intersection of two lines:
y = mx + b and x = a

1. The x-coordinate of the point is x = a.
2. Substitute x = a into y = mx + b and solve for
y. This is the y-coordinate of the point.


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Example Solve y = mx + b & x = a
Find the point of intersection of the lines
y = 2x - 1 and x = 2.
The x-coordinate of the point is x = 2. y
Substitute x = 2 into y = 2x - 1
to get the y-coordinate.
y = 2(2) - 1 = 3
Intersection Point: (2,3)
y = 2x - 1

Slide 44

(2,3)
x

x=2
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Supply Curve
p

For every quantity q of a
commodity, the supply
curve specifies the price
p that must be charged
for a manufacturer to be
willing to produce q units
of the commodity.

q

Supply Curve


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Demand Curve
p

For every quantity q of a
commodity, the demand
curve gives the price p
that must be charged in
order for q units of the
commodity to be sold.

q

Demand Curve


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Example Supply = Demand
Suppose the supply and demand for a quantity is
given by p = 0.0002q + 2 (p in dollars) and p = 0.0005q + 5.5. Determine both the quantity of
the commodity that will be produced and the
price at which it will sell when supply equals
demand.
p .0002q 2
.0005q 5.5
.0007q 3.5
q 5000 units
p .0002(5000) 2 $3

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Summary Section 1.3
 The point of intersection of a pair of lines can
be obtained by first converting the equations to
standard form and then either equating the two
expressions for y or substituting the value of x
from the form x = a into the other equation.


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Section 1.4

The Slope of a Straight
Line


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Outline
1. Slope of y = mx + b
2. Geometric Definition of
Slope
3. Steepness Property
4. Point-Slope Formula
5. Perpendicular Property
6. 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 that passes
through (-1,4) with a slope of 3 .
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
Find the equation of the line through the point
(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
Find the equation of the line through the point
(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
y = 2x - 11

2x + 4y = 7

y = (-1/2)x - 7/2

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 The slope of the line y = mx + b is the number
m. It is also the ratio of the difference between
the y-coordinates and the difference between the
x-coordinates of any pair of points on the line.
The steepness property states that if we start at
any point on a line of slope m and move 1 unit to
the right, then we must move m units vertically
to return to the line.

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 The point-slope formula states that the line of
slope m passing through the point (x1, y1) has the
equation y - y1 = m(x - x1).
 Two lines are parallel if and only if they have
the same slope. Two lines are perpendicular if
and only if the product of their slopes is –1.


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Section 1.5

The Method of Least
Squares


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Outline
1.
2.
3.
4.

Least Squares Problem
Least Squares Error
Least Squares Line
Least Squares Using
Technology


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Least Squares Problem
Least Squares Problem Given observed data
points (x1, y1), (x2, y2),…, (xN, yN) in the plane,
find the straight line that “best” fits these points.


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Least Squares Error
Least Squares Error
Let Ei be the vertical
distance between the point (xi, yi) and the straight
line. The least-squares error of the observed
points with respect to this line is
E = E12 + E22 +…+ EN2.


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Example Least Squares Error
Determine the least-squares error when the line
y = 1.5x + 3 is used to approximate the data
points (1,6), (4,5) and (6,14).
Vertical Distance

Ei2

(1, 4.5)

1.5

2.25

(4,9)

4

16

(6,12)

2

4

Data Point Point on Line

(1,6)
(4,5)
(6,14)

E = 22.25

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Graph of Least Squares Error
(6,14)
E3

(1,6)
E1

E2

(4,5)


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Least Squares Line
Least Squares Line Given observed data
points (x1, y1), (x2, y2),…, (xN, yN) in the plane,
the straight line y = mx + b for which the error E
is as small as possible is determined by
m

N

xy

N
b

x

y m
N

x
2

y
x

x

2

.


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Example Least Squares Error
Find the least-squares line for the data points
(1,6), (4,5) and (6,14).
x

y

xy

x2

1

6

6

1

4

5

20

16

6

14

84

36

x = 11

y = 25

xy = 110

x2 = 53


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Example Least Squares Error (2)
3 110 11 25 55
m
1.45
2
3 53 11
38
25 55 11
38
b
3.03
3
y 1.45 x 3.03


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Least Squares Using Technology
Use Excel to find the least-squares line for the
data points (1,6), (4,5) and (6,14).
y = 1.4474x + 3.0263


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Summary Section 1.5
 The method of least squares finds the straight
line that gives the best fit to a collection of points
in the sense that the sum of the squares of the
vertical distances from the points to the line is as
small as possible. The slope and y-intercept of
the least-squares line are usually found with
formulae involving sums of coordinates or by
using technology.


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Chapter 1 linear equations and straight lines

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