Transcript Chapter 3

Chapter 3
Linear Equations and
Functions
TSWBAT find solutions of two variable
open sentences, and graph linear
equations and points in two variables.
Two Variable Equations
 Open Sentence – a statement or
equation that can have more than one
solution.
 Open Sentence in Two Variables – A
statement that can have more than one
solution but contains two different
variables.
Two Variable Equations
 Solution Set – Set of all solutions to a given
problem.
 Solution – To a two variable sentence is an
Ordered Pair.
 Ordered Pair – Solution to a two variable
sentence or the value of the x and y terms to a
point on a graph. An ordered pair is written
with the x-term first and the y-term second.
EX (X, Y); (2,5); (-5,3); (8,-1)
Graphing
 xy- Coordinate Plane – The normal two
directional plane on which an open
sentence in two variables can be
graphed.
Graphing
 Plane Rectangular Coordinate System
or Cartesian coordinate system– Made
up of two number lines that intersect at
right angles at the point O. This is also
named the Cartesian system after French
mathematician Rene Descartes who
introduced the idea of coordinates.
Graphing
 Origin – The intersection point of the two
number lines in the coordinate system
labeled as point O.
 x-axis – The horizontal number line in a
Coordinate System.
 y-axis – The vertical number line in a
Coordinate System.
Graphing
 Quadrants – The four sections made up by the
intersection of the x and y axis in the coordinate
system.
 Plotting – In graphing points and lines on the
coordinate system we call graphing a point plotting the
point or placing the point on the graph.
 Domain – The values for which x can be in a two
variable sentence or on the coordinate system.
 Range – The values for which y can be in a two
variable sentence or on the coordinate system.
Graphing
 One – to – One Correspondence –
between ordered pairs and points on the
plane can be summarized as:
 1. There is exactly one point on the plane
associated with each ordered pair.
 2. There is exactly one ordered pair
associated with each point on the plane.
Graphing
 Graph - of an open sentence in two variables
is the set of all points in the coordinate plane
that satisfies the sentence.
 Linear Theorem - The graph of every
equation of the form : Ax + By = C, when A
and B are not both 0, is a line. Similarly every
line in the coordinate plane is the graph of an
equation in this form.
Linear vs. Non-Linear
 Linear – Forms a line
 Examples of linear: 5x + 3y = -8, x  y  2
5
 Non-Linear – Does not form a line.
 Examples of Non-linear: 2x +3y2 = 4,
xy = 2,
3
2x 
y
5
5
X and Y Intercepts
 To solve for x-Intercept
1. Solve equation for X.
2. Substitute 0 in for y.
3. Solve
 To solve for y-Intercept
1. Solve equation for y.
2. Substitute 0 in for x.
3. Solve
Graph a Line
 It is best to have 3 points on the line, but
you only need 2.
 The easiest way is to graph the two
intercepts and then plot the third point
you are given or find to determine the
direction of the line.
Examples
 Finding Solutions to two variable
equations
Graphing
 Graphing Points and Lines
Graphing
 Finding X and Y Intercepts
Chapter 3
TSWBAT Find Slope of a line, and
graph a line given the slope and
point on the line.
Slope
rise y 2  y1

 Slope of a Line L = run x 2  x1
where ( x  x )
2
1
 Horizontal Line – Slope = 0
 Vertical Line – No Slope
 Coefficient – number or numerical factor
in front of a variable. In the y-equals
equation the coefficient in front of the xterm is the slope of the line.
Slope Theorems
 Theorem 2 – The slope of the line
A
(
B

0
)
Ax + By = C where
is  .
B
 Theorem 3 – Let P(x1,y1) be a point and
m a real number. There is one and only
one line L through P having the slope m.
An equation of L is y – y1 = m(x – x1).
Slope Generalizations
 The slope of a line rises if m is positive.
 The slope of a line falls if m is negative.
 The larger m is, the steeper the line is.
Slope
 Examples – Finding Slope
Chapter 3
TSWBAT find an equation of a line given
the slope and a point on the line, given
two points, or given the slope and yintercept.
Equations for a Line
 Standard Form of the equation of a line is
Ax + By = C with A, B, and C being integers.
 There are two other forms for the equation of a
line however.
 Point-Slope form – the equation is then
y – y1 = m(x – x1).
 Slope-Intercept form – the equation is
y = mx + b.
Finding Equation of a Line
 Examples – Standard Form
Finding Equation of a Line
 Examples – Point Slope Form
Finding Equation of a Line
 Examples – Slope Intercept Form
Chapter 3
TSWBAT find equations of parallel and
perpendicular lines, find linear functions and graph
them and determine if relations are functions.
Parallel Lines
 Parallel Lines have the same slope and
never intersect.
 Example
Perpendicular Lines
 Have the opposite reciprocal slope of the
other line.
 These lines meet at only one point in a
90 degree angle.
 Example
Functions and Relations
 Function – a correspondence between
two sets, D and R, that assigns to each
member of D exactly one member of R.
(One to One Correspondence).
 Example:
 Domain of the Function – is the Set D.
 Example:
 Range of the Function – is the Set R.
 Example:
Functions and Relations
 Values of a function – the members of the
range assigned to a member of the domain.
 Example: the function f assigns 2 the value 4.
 Functional notation – f(x)=C Example: f(2)=4
 Linear Functions – A function f that can be
defined by the equation f(x)=mx+b where x, m,
and b are real numbers, and the graph of f is
the graph of the line y=mx+b with slope m and
y-intercept b.
 Example:
Functions and Relations
 Constant Function – A function where
m=0 and is thus f(x)=b for all x.
 Example:
 Is this a Horizontal or Vertical Line?
 Rate of change m = slope of a line =
change in f ( x)
change in x
Functions and Relations
 Relation – Any set of ordered pairs. A
function is a relation but not all
relations are functions. A relation can
contain two or more ordered pairs
with the same x and/or y values. A
function can contain two or more
ordered pairs with the same y value
only.
 Example:
Functions and Relations
 Vertical - Line Test – a test to
determine if a given relation is a
function. This test says a relation is a
function if and only if a vertical line
intersects the graph of the relation at
most one time.
 Example:
Chapter 3
TSWBAT Solve systems of Linear
Equations by 1. Linear Combinations, 2.
Substitution, 3 Graphing.
Systems of Linear
Equations
 A system of linear equations or linear
system –
a set of linear equations in the
same two variables.
Example
Solutions to a Systems of
Linear Equations
 1. simultaneous solution - an ordered
pair that satisfies both equations at
their point of intersection.
 Example
Solutions to a Systems of
Linear Equations
 2. the null set for two lines that are
parallel.
 Example
Solutions to a Systems of
Linear Equations
 3. a line if the set of linear systems is
a group of coinciding lines.
 Example
Systems of Linear
Equations
 Equivalent systems – systems of
linear equations that have the same
solution set.
 Example
 Linear Combination – the addition of
two equations.
 Example
Systems of Linear
Equations
 Consistent system – a system that has
at least one solution.
 Example
 Inconsistent system – a system with
no solution and lines that are
inconsistent.
 Example
Systems of Linear
Equations
 Dependent system – a system that has
an infinite number of solutions and the
lines are coinciding.
 Example
Transformations
 1.Replacing an equation by an
equivalent expression. – That is
multiplying each side of an equation
by the same non-zero number.
Transformations

2. Substituting for one variable in
an equation for that variable obtained
from another equation in the system.
Transformations

3. Replacing any equation by the
sum of that equation and another
equation in the system. – That is add
left sides, right sides, and then equate
the results.
Systems of Linear
Equations
 Three methods to solve
 1. Linear Combination
 2. Substitution
 3. Graphing
Systems of Linear
Equations
 Example Linear Combination
Systems of Linear
Equations
 Example Substitution
Systems of Linear
Equations
 Example Graphing
Chapter 3
TSWBAT solve linear inequalities
and systems of linear inequalities.
Linear Inequalities
 Linear Inequality in Two Variables – is
when the equals sign in a linear equation
in two variables is replaced by an
inequality symbol like <, >, , or  .
 Boundary – The linear equation from
which the inequality was formed.
 Solution – a shaded region defined by
the inequality symbol and boundary.
Linear Inequalities
 Open Half-Plane – When the boundary
line is not included in the solution and is
shown as a dashed line (when we have <
or >).
 Closed Half-Plane – When the boundary
line is included in the solution and is
shown as a solid line (when we have 
or  ).
Linear Inequalities
 Example -
System of Linear
Inequalities
 System of Inequalities – Two or more
linear inequalities working together as a
set.
 Solution to a system of Inequalities – is
the region where ALL inequalities have a
shaded region as a solution.
System of Linear
Inequalities
 Example