Chapter 4 Linear Equations and Inequalities in Two Variables

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# Chapter 4 Linear Equations and Inequalities in Two Variables - PowerPoint PPT Presentation

Chapter 4 Linear Equations and Inequalities in Two Variables. § 4.1. Graphing Linear Equations in Two Variables. The Rectangular Coordinate System. In the rectangular coordinate system, the horizontal number line is the x-axis .

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§4.1

Graphing Linear Equations in Two Variables

The Rectangular Coordinate System

In the rectangular coordinate system, the horizontal number line is the x-axis.

The vertical number line is the y-axis. The point of intersection of these axes is

their zero points, called the origin. The axes divide the plane into 4 quarters, called quadrants.

y-axis

x-axis

Blitzer, Introductory Algebra, 5e – Slide #3 Section 4.1

The Rectangular Coordinate System

Each point in the rectangular coordinate system corresponds to an ordered pair of real numbers (x,y). Note the word “ordered” because order matters. The first number in each pair, called the x-coordinate, denotes the distance and direction from the origin along the x-axis. The second number, the y-coordinate, denotes vertical distance and direction along a line parallel to the y-axis or along the y-axis Itself.

In plotting points,

we move across first (either left or right), and

then move either up or

down, alwaysstarting from the origin.

Blitzer, Introductory Algebra, 5e – Slide #4 Section 4.1

Plotting Points

EXAMPLE

Plot the points (3,2) and (-2,-4).

SOLUTION

(3,2)

(-2,-4)

Blitzer, Introductory Algebra, 5e – Slide #5 Section 4.1

The Graph of an Equation

The graph of an equation in two variables is the set of

points whose coordinates satisfy the equation.

An ordered pair of real numbers (x,y)

is said to satisfy the equation when substitution of the x and y coordinates

into the equation makes it a true statement.

For example, in the equation

y = 2x + 6, the ordered pair (1,8) is a solution. When we substitute this point the sentence reads 8 = 8, which is true.

The ordered pair (2,3) is not a solution. When we substitute this point, the sentence reads 3 = 10, which is not true.

Blitzer, Introductory Algebra, 5e – Slide #6 Section 4.1

Solution of an Equation in Two Variables

Remember….

If when the x-coordinate of the ordered pair is substituted for x and the y-coordinate of the ordered pair is substituted for y, we obtain a true statement- then the ordered pair is a solution to the equation.

Blitzer, Introductory Algebra, 5e – Slide #7 Section 4.1

Solution of an Equation in Two Variables

Example:

Given the equation 2x + 3y = 18, determine if the ordered pair (3, 4) is a solution to the equation.

We substitute 3 in for x and 4 in for y.

2(3) + 3 (4) ? 18

6 + 12 ? 18

18 = 18 True.

Therefore, the ordered pair (3, 4) is a solution to the equation 2x + 3y = 18.

Blitzer, Introductory Algebra, 5e – Slide #8 Section 4.1

Finding Solutions of an Equation

Find five solutions to the equation y = 3x + 1.

Start by choosing some x values and then computing the corresponding y values.

If x = -2, y = 3(-2) + 1 = -5. Ordered pair (-2, -5)

If x = -1, y = 3(-1) + 1 = -2. Ordered pair ( -1, -2)

If x =0, y = 3(0) + 1 = 1. Ordered pair (0, 1)

If x =1, y = 3(1) + 1 =4. Ordered pair (1, 4)

If x =2, y = 3(2) + 1 =7. Ordered pair (2, 7)

Blitzer, Introductory Algebra, 5e – Slide #9 Section 4.1

Graph of the Equation

Plot the five ordered pairs to obtain the graph of y = 3x + 1

(2,7)

(1,4)

(0,1)

(-1,-2)

(-2,-5)

Blitzer, Introductory Algebra, 5e – Slide #10 Section 4.1

Graphing an Equation

EXAMPLE

Graphs in the rectangular coordinate system can also be used to tell a story. Try to select the graph that best illustrates the story of the population of the U.S.A.

(a)

(b)

(c)

Population

Population

Population

Years

Years

Years

SOLUTION

Graph (c)

Blitzer, Introductory Algebra, 5e – Slide #11 Section 4.1

Linear Equations

Definition of a Linear Equation

A linear equation in one variable x is an equation that can be written in the form ax + b = 0, where a and b are real numbers and a is not equal to 0.

An example of a linear equation in x is 4x + 2 = 6. Linear equations in x are first degree equations in the variable x.

Blitzer, Introductory Algebra, 5e – Slide #12 Section 4.1

Comparison of Graphs of Linear Equations

The graph of y = mx + b is the graph of

y = mx shifted b units up when b is a positive number.

The graph of y = mx + b is the graph of

y = mx shifted b units down when b is a negative number.

More on lines to come in this chapter.…

Blitzer, Introductory Algebra, 5e – Slide #13 Section 4.1