Section 4.1. Points and Lines. Math 081

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Math 081

Section 4.1

Points and Lines

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D ^IRECTIONS

As you work through this Learning Guide, you should:

 read carefully

 take notes

 do the PRACTICE problems on your own

 check your answers to the PRACTICE problems

 watch the videos by clicking on the icons

IMPORTANT: Get help if you don’t understand a topic.

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Q

UICK

L

INKS

:

Use the links below to go directly to a topic.

 Introduction and Coordinate Plane

 Points on the Coordinate Plane

 Ordered Pairs for Points

 Plotting a Point

 Graph of a Line

 Equation of a Line

 Solution of a Linear Equation in Two Variables

 Determining an Unknown Coordinate in a Solution

 Organizing Solutions of Equations Using Tables

 Intercepts

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I

NTRODUCTION

A

^ND

C

^OORDINATE

P

^LANE

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A coordinate plane is formed by two number lines, a horizontal number line and a vertical number line, that intersect at a point called the origin.

The number lines are called axes.

The horizontal number line is the x-axis and the vertical number line is the y-axis.

These two number lines divide the plane (a

two-dimensional surface) into four quadrants.

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P

OINTS ON THE

C

^OORDINATE

P

^LANE

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O

^RDERED

P

^{AIRS FOR}

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^OINTS

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Now you will learn how to determine the ordered pair for a point that is plotted on the coordinate plane.

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EXAMPLE: Write the ordered pair for the point.

1.

( , )x y

So point M is represented by the ordered pair .(2,3) Start at the origin.

Since we moved 2 units to the right, the x-coordinate is 2.

From our current location, we need to move 3 units up to get to point M.

The y-coordinate is 3.

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PRACTICE: Write the ordered pair for each point.

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( , )x y

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PRACTICE: Answers Write the ordered pair for each point.

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A (3, 4)

( , )x y

B (0, 2)

C ( 5, 4) 

D ( 3,0) 

E ( 4, 5)  

F (2, 4) 

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P

^{LOTTING A}

P

^OINT

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EXAMPLE: Plot the point B . 1.

( 4, 2)

The x-coordinate indicates how far to move left or right from

the origin.

The x-coordinate is – 4.

Since it is negative, we move 4 units to the left.

The y-coordinate indicates how far to move up or down from

our current location.

The y-coordinate is 2

Since it is positive, we move 2 units up.

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PRACTICE: Plot the following points and label each using the capital letter.

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PRACTICE: Answers Plot the following points and label each using the capital letter.

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G

^{RAPH OF A}

L

^INE

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The graph of a line is a drawing of the line on the coordinate plane.

A line is perfectly straight.

A line may be positioned in any direction: horizontal, vertical, slanted upwards, or slanted downwards.

Arrows are placed on a line to show that the line extends infinitely in both directions.

Note: You should be aware that the graphs of lines shown may not display the arrows due to a limitation of the technology used to draw the lines. Despite this, you should remember that all lines are endless.

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E

QUATION OF A

L

^INE

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There is a relationship between the x-coordinates and the y-coordinates of all the points on a particular line.

That relationship can be expressed as an algebraic equation.

Equations whose graphs are lines are called linear equations in two variables.

Often the equation representing a line will be given in the form Ax + By = C where A, B, and C are real numbers.

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S

OLUTION OF A

L

^INEAR

E

^{QUATION IN}

T

^WO

V

^ARIABLES

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EXAMPLE: Determine if the given ordered pair is a solution of the equation.

1.

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PRACTICE: Determine if the ordered pair is a solution of the equation.

1.

2.

3.

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PRACTICE: Answers Determine if the ordered pair is a solution of the equation.

1.

2.

3.

20

Yes

No

Yes

No

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D

ETERMINING AN

U

^NKNOWN

C

OORDINATE IN A

S

^OLUTION

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EXAMPLE: Determine the unknown coordinate so that the ordered pair is a solution of the equation.

1.

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EXAMPLE: Determine the unknown coordinate so that the ordered pair is a solution of the equation.

2.

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PRACTICE: Determine the unknown coordinate so that the ordered pair is a solution of the equation.

1.

2.

3.

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PRACTICE: Answers Determine the unknown coordinate so that the ordered pair is a solution of the equation.

1.

2.

3.

25

(2, 1) 

(0,7)

(4,5)

( 4,1) 

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O

^RGANIZING

S

OLUTIONS OF

E

^QUATIONS

U

^SING

T

^ABLES

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Remember that a line is made up of an infinite number of points. Therefore, the equation of a line has an infinite number of solutions. We cannot list all the solutions, but we can find and list some of the solutions.

One way to organize ordered pair solutions of an equation is to use a table format.

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EXAMPLE: Complete the table to find several solutions of the equation . 1.

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5 x y

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PRACTICE: Complete the table to find solutions of the given equation.

1.

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4x y 4

  

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PRACTICE: Answers Complete the table to find solutions of the given equation.

1.

29

4x y 4

  

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I

^NTERCEPTS

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There are two solutions of a linear equation that are very important. One is an ordered pair with an x-coordinate of 0, and the other is an ordered pair with a y-coordinate of 0.

These two solutions are important because they correspond to the points where the line crosses the axes of the coordinate plane.

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EXAMPLE: Determine the x and/or y intercept as specified.

1.

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EXAMPLE: Determine the x and/or y intercept as specified.

2.

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PRACTICE: Determine the x and/or y intercept as specified.

1.

2.

3.

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PRACTICE: Answers Determine the x and/or y intercept as specified.

1.

2.

3.

34

3 x 

10 y 

9 and 3

x  y 

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35

Section 4.1. Points and Lines. Math 081

Math 081

Section 4.1

Points and Lines

D IRECTIONS

As you work through this Learning Guide, you should:

 read carefully

 take notes

 do the PRACTICE problems on your own

 check your answers to the PRACTICE problems

 watch the videos by clicking on the icons

IMPORTANT: Get help if you don’t understand a topic.

Q

L

:

Use the links below to go directly to a topic.

 Introduction and Coordinate Plane

 Points on the Coordinate Plane

 Ordered Pairs for Points

 Plotting a Point

 Graph of a Line

 Equation of a Line

 Solution of a Linear Equation in Two Variables

 Determining an Unknown Coordinate in a Solution

 Organizing Solutions of Equations Using Tables

 Intercepts

I

A

C

P

A coordinate plane is formed by two number lines, a horizontal number line and a vertical number line, that intersect at a point called the origin.

The number lines are called axes.

The horizontal number line is the x-axis and the vertical number line is the y-axis.

These two number lines divide the plane (a

two-dimensional surface) into four quadrants.

P

C

P

O

P

P

A (3, 4)

B (0, 2)

C ( 5, 4) 

D ( 3,0) 

E ( 4, 5)  

F (2, 4) 

P

P

G

L

E

L

S

L

E

T

V

Yes

No

Yes

Yes

No

D

U

C

S

(2, 1) 

(0,7)

(4,5)

( 4,1) 

O

S

E

U

T

I

3 x 

10 y 

9 and 3

D ^IRECTIONS