CHAPTER 3

Linear Equations and Functions

SECTION 3-1

Open Sentences in Two Variables

Open sentences in two variables– equations and inequalities containing two variables

DEFINITIONS

9x + 2y = 15

y = x2 – 4

2x – y ≥ 6

EXAMPLES

Solution– is a pair of numbers (x, y) called an ordered pair.

DEFINITIONS

Solution set– is the set of all solutions satisfying the sentence. Finding the solution is called solving the open sentence.

DEFINITIONS

Solve the equation: 9x + 2y = 15

if the domain of x is {-1,0,1,2}

EXAMPLE

SOLUTION

x (15-9x)/2 y Solution

-1 [15-9(-1)]/2 12 (-1,12)

0 [15-9(0)]/2 15/2 (0,15/2)

1 [15-9(1)]/2 3 (1,3)

2 [15-9(2)]/2 -3/2 (2,-3/2)

the solution set is{(-1,12), (0, 15/2), (1,3), (2,-3/2)}

SOLUTION

Roberto has $22. He buys some notebooks costing $2 each and some binders costing $5 each. If Roberto spends all $22 how many of each does he buy?

EXAMPLE

n = number of notebooksb = number of binders(n and b must be whole

numbers)2n + 5b =22n = (22-5b)/2

SOLUTION

If n is odd then 22-5b is odd and n is not a whole number

SOLUTION

SOLUTION

b (22-5b)/2 n Solution

0 [22-5(0)]/2 11 (0,11)

2 [22-5(2)]/2 6 (2,6)

4 [22-5(4)]/2 1 (4,1)

6 [12-5(6)]/2 -4 Impossible

the solution set is{(0,11), (2, 6), (4,1)}

SOLUTION

SECTION 3-2

Graphs of Linear Equations in Two Variables

COORDINATE PLANE consists of two

perpendicular number lines, dividing the plane into four regions called

quadrants

COORDINATE PLANE

X-coordinate (abscissa)- the horizontal number line

Y-coordinate (ordinate) - the vertical number line

ORIGIN - the point where the x-coordinate and

y-coordinate cross

ORDERED PAIR - a unique assignment of real

numbers to a point in the coordinate plane

consisting of one x-coordinate and one y-

coordinate(-3, 5), (2,4), (6,0), (0,-3)

DEFINITION

1.There is exactly one point in the coordinate plane associated with each ordered pair of real numbers.

ONE-TO-ONE CORRESPONDENCE

2. There is exactly one ordered pair of real numbers associated with each point in the coordinate plane.

ONE-TO-ONE CORRESPONDENCE

GRAPH – is the set of all points in the coordinate plane whose coordinates

satisfy the open sentence.

DEFINITION

The graph of every equation of the form

Ax + By = C (A and B not both zero) is a line. Conversely, every line in the coordinate

plane is the graph of an equation of this form

THEOREM

LINEAR EQUATIONis an equation whose

graph is a straight line.

SECTION 3-3

The Slope of a Line

SLOPE

is the ratio of vertical change to the horizontal

change. The variable m is used to represent slope.

m = change in y-coordinate change in x-coordinate

Or m = rise

run

FORMULA FOR SLOPE

SLOPE OF A LINEm = y2 – y1

x2 – x1

HORIZONTAL LINE

a horizontal line containing the point

(a, b) is described by the equation y = b and has slope

of 0

VERTICAL LINE

a vertical line containing the point (c, d) is described by

the equation x = c and has no slope

Find the slope of the line that contains the given points.

M(4, -6) and N(-2, 3)

The slope of the line Ax + By = C (B ≠ 0) is

- A/B

THEOREM

Let P(x1,y1) be a point and m a real number. There is one and

only one line L through P having slope m. An equation

of L is

y – y1 = m (x – x1)

THEOREM

Write an equation of a line with the given slope and through a given point

m= -2P(-1, 3)

SECTION 3-4

Finding an Equation of a Line

POINT-SLOPE FORM

y – y1 = m (x – x1)

where m is the slope and (x1

,y1) is a point on the line.

Write an equation of a line with the given slope and passing through a given point in standard form

m= -2P(-1, 3)

Y-Intercept

is the point where the line intersects the y -

axis.

X-Intercept

is the point where the line intersects the

x -axis.

SLOPE-INTERCEPT FORM

y = mx + bwhere m is the slope and b

is the y -intercept

Write an equation of a line with the given y-intercept and slope

m=3 b = 6

Let L1 and L2 be two different lines, with slopes m1 and m2 respectively.

1. L1 and L2 are parallel if and only if m1=m2

THEOREM

and2. L1 and L2 are

perpendicular if and only if m1m2 = -1

THEOREM

Write an equation of a line passing through the given points

A(1, -3) B(3,2)

Find the slope of a line parallel to the line containing points M and N.

M(-2, 5) and N(0, -1)

Find the slope of a line perpendicular to the line

containing points M and N.

M(4, -1) and N(-5, -2)

Write an equation of a line parallel to y=-1/3x+1 containing the point (1,1)

m=-1/3

Write an equation of a line perpendicular to y= -1/3x+1 containing the point (1,1)

m=-1/3P(1, 1)

SECTION 3-5

Systems of Linear Equations in Two Variables

SYSTEM OF EQUATIONS

Two linear equations with the same two variable form

a system of equations.

SOLUTION

The ordered pair that makes both equations true.

The SOLUTION to the system of equations is

a point (point of intersection of the two

lines).

INTERSECTING LINES

There is NO SOLUTION to the system of

equations (no intersection of the two

lines).

PARALLEL LINES

The graph of each equation is the same. The lines coincide and

any point on the line is a solution.

COINCIDING LINES

EQUIVALENT SYSTEMS

Systems that have the same solution.

METHODS FOR SOLVING SYSTEM OF EQUATIONS

GraphingSubstitutionLinear-Combination

SOLVE BY GRAPHING

4x + 2y = 8

3y = -6x + 12

SOLVE BY GRAPHING

y = 1/2x + 3

2y = x - 2

SUBSTITUTION

A method for solving a system of equations by

solving for one variable in terms of the other variable.

SOLVE BY SUBSTITUTION

3x – y = 6x + 2y = 2

Solve for y in terms of x.3x – y = 63x = 6 + y

3x – 6 = y then

SOLVE BY SUBSTITUTION

Substitute the value of y into the second equation

x + 2y = 2x + 2(3x – 6) = 2x + 6x – 12 = 2

7x = 14x = 2 now

SOLVE BY SUBSTITUTION

Substitute the value of x into the first equation

3x – y = 6y = 3x – 6

y = 3(2 – 6)y = 3(-4)y = -12

SOLVE BY SUBSTITUTION

2x + y = 0x – 5y = -11

Solve for y in terms of x.2x + y = 0

y = -2xthen

SOLVE BY SUBSTITUTION

Substitute the value of y into the second equation

x – 5y = -11x – 5(-2x) = -11x+ 10x = -11

11x = -11x = -1

SOLVE BY SUBSTITUTION

Substitute the value of x into the first equation

2x + y = 0y = -2x

y = -2(-1)y = 2

LINEAR-COMBINATION

Another method for solving a system of equations where one of the variables is eliminated by adding or subtracting the two

equations.

If the coefficients of one of the variables are opposites, add the equations to eliminate one of the variables. If the coefficients of one of the variables are the same, subtract the equations to eliminate one of the variables.

LINEAR COMBINATION

Solve the resulting equation for the remaining variable.

LINEAR COMBINATION

Substitute the value for the variable in one of the original equations and solve for the

unknown variable.

LINEAR COMBINATION

Check the solution in both of the original equations.

LINEAR COMBINATION

This method combines the multiplication property of

equations with the addition/subtraction method.

LINEAR COMBINATION

3x – 4y = 103y = 2x – 7

SOLVE BY LINEAR-COMBINATION

SOLUTION

3x – 4y = 10-2x +3y = -7

Multiply equation 1 by 2Multiply equation 2 by 3

SOLUTION

6x – 8y = 20-6x +9y = -21

Add the two equations.

y = -1

SOLUTION

Substitute the value of y into either equation and solve for

3x – 4y = 103x – 4(-1) = 10

3x + 4 = 103x = 6x = 2

CONSISTENT SYSTEM

The system of equations has at least one solution.

INCONSISTENT SYSTEM

The system of equations has no solution.

DEPENDENT SYSTEM

The graph of each equation is the same. The lines

coincide and any point on the line is a solution.

SECTION 3-6

Problem Solving: Using Systems

If 8 pens and 7 pencils cost $3.37 while 5 pens and 11 pencils cost $3.10, how much does each pen and each pencil cost?

EXAMPLE

Let x = cost of a peny = cost of a pencil8x + 7y = 3.375x + 11y = 3.10

SOLUTION

Solve the system of equations.

x = 29y = 15

SOLUTION

To use a certain computer data base, the charge is $30/hr during the day and $10.50/hr at night. If a research company paid $411 for 28 hr of use, find the number of hours charged at the daytime rate and at the nighttime rate.

EXAMPLE

Let x = number of hrs at daytime rate

y = number of hrs at nighttime rate

30x + 10.50y = 411x + y = 28

SOLUTION

Solve the system of equations.

x = 6y = 22

SOLUTION

SECTION 3-7

Linear Inequalities in Two Variables

Linear equations that have the equal sign replaced by

one of these symbols <, ≤, ≥, >

SYSTEM OF INEQUALITIES

SYSTEM OF LINEAR INEQUALITIES

The SOLUTION of an inequality in two variables is an ordered pair of numbers that satisfies

the inequality.

SYSTEM OF LINEAR INEQUALITIES

A system of linear inequalities can be solved by graphing each

associated equation and determining the region where

the inequality is true.

HALF-PLANE

Graphically its the region on either side of the line.

BOUNDARY

is the line separating the half-planes

OPEN HALF-PLANE

is the region on either side of the boundary line (illustrated by a dashed-

line).

CLOSED HALF-PLANE

is the solution that includes the boundary

line.

is a graph of all the solutions of the inequality and includes a boundary,

either a solid line or dashed line and a shaded

area.

GRAPH of an INEQUALITY

is a point that does not lie on the boundary, but

rather above or below it.

TEST POINT

x + y ≥ 4

(0,4),(4,0)

GRAPHING INEQUALITIES

Shade the region above a dashed line if y > mx +

b.

Shade the region above a solid line if

y mx + b.

Shade the region below a dashed line if y < mx +

b.

Shade the region below a solid line if

y ≤ mx + b.

SOLVE BY GRAPHING THE INEQUALITIES

x + 2y < 52x – 3y ≤ 1

SOLVE BY GRAPHING THE INEQUALITIES

4x - y 58x + 5y ≤ 3

SECTION 3-8

Functions

MAPPING DIAGRAM

A picture showing a correspondence between two sets

MAPPING – the relationship between the elements of the domain and range

DOMAIN – the set of all possible x-coordinates

RANGE – the set of all possible y-coordinates

FUNCTION

A correspondence between two sets, D and R, that assigns to each member of D exactly one member of R.

Introduction to Functions

Definition – A function f from a set D to a set R is a relation that assigns to each element x in the set D exactly one element y in the set R. The set D is the domain of the function f, and the set R contains the range

Characteristics of a Function

1. Each element in D must be matched with an element in R.

2. Some elements in R may not be matched with any element in D.

3. Two or more elements in D may be matched with the same element in R.

4. An element in D (domain) cannot be match with two different elements in R.

Example

A = {1,2,3,4,5,6} and B = {9,10,12,13,15}

Is the set of ordered pairs a function?

{(1,9), (2,13), (3,15), (4,15), (5,12), (6,10)}

Given f: x→4x – x2 with domain D= {1,2,3,4,5}

Find the range of f.f, the function that assigns to x the number 4x – x2

EXAMPLE

SOLUTION

x 4x-x2 (x,y)1 4(1) – 1 = 3 (1,3)2 4(2) – 4 = 4 (2,4)3 4(3) – 9 = 3 (3,3)4 4(4) – 16 = 0 (4,0)5 4(5) – 25 = -5 (5,-5)

FUNCTIONAL NOTATION

f(x) denotes the value of f at x

VALUES of a FUNCTION

The members of its range.

SECTION 3-9

Linear Functions

Linear FunctionIs a function f that can be

defined by f(x) = mx + bWhere x, m and b are real

numbers. The graph of f is the graph of y = mx +b, a line with slope m and y -intercept b.

Constant Function

If f(x) = mx + b and m = 0, then f(x) = b for all x and its graph is a horizontal line y = b

Rate of Change m

Rate of Change m =

change in f(x)Change in x

EXAMPLE

Find equations of the linear function f using the given information.

f(4) = 1 and f(8) = 7

SOLUTION

m = f(8) – f(4) 8 – 4m = 3/2

SECTION 3-10

Relations

RELATION

Is any set of ordered pairs. The set of first coordinates in the ordered pairs is the domain of the relation, and

RELATION

and the set of second coordinates is the range.

FUNCTION

is a relation in which different ordered pairs have different first coordinates.

VERTICAL LINE TEST

a relation is a function if and only if no vertical line intersects its graph more than once.

Determine if Relation is a Function

{2,1),(1,-2), (1,2)}

{(x,y): x + y = 3}

END