Graphing Linear Equations

X and Y Coordinates

A coordinate or ordered-pair notation is always written as (x-coordinate, y-coordinate) or (x, y).

They are always in parentheses with the value of x always first and the value of y always second. That will never change.

Coordinates allow us to identify a specific place on a graphing system called the rectangular coordinate system.

A linear equation in two variables looks like:2x + y = 9 because it contains two variables, namely x and y.

Solutions of Equations in Two Variables

Determine if the following ordered pairs are a solution of the equation: x – 2y = 6

(6, 0) (0, 3) (1, -5/2)

Substitute the ordered pairs into the given equation:

6 – 2(0) = 6 0 – 2(3) = 6 1 – 2(-5/2) = 6

6 – 0 = 6 -6 = 6 1 + 5 = 6

6 = 6 -6 = 6 6 = 6

yes no yes

You try it….

Determine which of the ordered pairs are solutions for the given equation:

2x – 3y = 6 (0, 2) (3, 0), (6, 2) (0, -2)

Do the work first and then click the mouse button to see if you got them right!

Did you get them right?

2(0) – 3(2) = 6

2 – 6 = 6

-4 = 6

No

2(3) – 3(0) = 6

6 – 0 = 6

6 = 6

Yes

2(6) – 3(2) = 6

12 – 6 = 6

6 = 6

Yes

2(0) – 3(-2) = 6

0 + 6 = 6

6 = 6

Yes

What this tells us is that the coordinates (3, 0), (6, 2), and (0, -2) can all be plotted on a graphing system and we can connect them together in a straight line. The point (0, 2) will not be on the line.

Completing Ordered Pairs

When finding a missing coordinate in a pair, you substitute the known coordinate into the equation and solve the equation to find the missing coordinate. For instance:

x + y = 12 (4, ), ( , 5), (0, ), ( , 0)

4 + y = 12

y = 8

x + 5 = 12

x = 7

0 + y = 12

y = 12

x + 0 = 12

x = 12

The complete coordinates are then:

(4, 8), (7, 5), (0, 12), and (12, 0)

You try it …..

5x – y = 15 ( ,0), (2, ), (4, ), ( , -5)

5x – 0 = 15

5x = 15

x = 3

5(2) – y = 15

10 – y = 15

-y = 5

y = -5

5(4) – y = 15

20 – y = 15

-y = -5

y = 5

5x – (-5) = 15

5x + 5 = 15

5x = 10

x = 2

The complete ordered pairs would be (3, 0), (2, -5), (4, 5), (2, -5).

How did you do? Keep practicing if you are having problems.

The Rectangular Coordinate System

Graphing

Rectangular Coordinate System

Quadrant 1

(+, +)

Quadrant 2

(-, +)

Quadrant 3

(-, -)

Quadrant 4

(+, -)

y-axis

x-axis•

Where the x-axis and y-axis meet is called the origin.

• (1,5)

The point (1, 5) is in line with the 1 of the x-axis and 5 on the

y-axis.

Completing a Table of Values

Complete the table for the equation 2x + y = 4

x y

0 4

2 0

? 2

Substitute the known value from the table into the equation and solve for the other variable.

2(0) + y = 4 2x + 0 = 4

0 + y = 4 2x = 4

y = 4 x = 2

What is the value of x when y = 2?

Click here to see if you were right.

Graphing Linear Equations

A linear equation in two variables is an equation that can be written in the form

ax + by = c (a, b, and c are coefficients)

Graph the linear equation x + y = 7

Create a Table of Values and choose numbers for x or y. In this example we will substitute values for x. You can substitute any value you want and can choose either the x or y value to assign numbers to, it’s your choice.

x y

0

1

2

Hint: It’s easiest to choose the numbers 0, 1, or 2 when filling in the table of values. If fractions are involved, then use

multiplies of the denominator.

Graphing Linear Equations (cont’d)

Complete a table of values for the equation x + y = 7

0 + y = 71 + y = 7 2 + y = 7y = 7 y = 6 y = 5

x y

0 7

1 6

2 5

The completed table of values will give us the coordinates that we will use to plot the line on the rectangular coordinate system.

(0, 7)

(1, 6)

(2, 5)

Graphing Linear Equations (cont’d)

x y

0 7

1 6

2 5 •(0, 7)•(1, 6)

•(2, 5)

(0, 7)

(1, 6)

(2, 5)

After you have found the points from the Table of

Values and you plot them on the graph, your points must

line up in a straight line. If they do not, then you made a mistake in your

math and need to go back and check to see what went

wrong.

Vertical and Horizontal Lines

Vertical lines have an equation of x=c.

Horizontal lines have an equation of y=c.

y = c

x = c

Examples of Horizontal and Vertical Lines

x = 5Since x is equal to a coefficient, in this case 5, then this is an equation of a vertical line.

x = 5

The red line means that

whenever y is equal to any

number, the x value will

always be 5. Examples of coordinates would be:

(5, 7), (5, 10), (5,-14), (5, -1)The value of x

is always 5 which means it is a vertical line.

Examples of Horizontal and Vertical Lines

y = 5Since y is equal to a coefficient, then this is an equation of a horizontal line.

The blue line means that

whenever x is equal to any

number, the y value will always

be 5. Examples of coordinates would be:

(-1, 5), (-8, 5), (7, 5), (100, 5)

The value of y is always 5 which means it is a

horizontal line.

y = 5

Slope of a Line

The formula for finding the slope of a line is:

1212

12 xxwherexx

yy

changehoriz

changevertm

),(),( 2211 yxandyx

Caution: Be careful when substituting in the values of x and y into the formula. You may want to label your x and y variables in the coordinates so you do not mix them up. The “sub exponent 1” and “sub exponent 2” are very important.

Example for Finding the Slope of a Line

Find the slope of the line with coordinates of (5, 7) and (9, 11).

)11,9()7,5(

),(),( 2211

and

yxandyx

14

4

59

711

12

12

xx

yym

11

7

9

5

2

1

2

1

y

y

x

x

Labeling your points is very important.

You try it …..

Find the slope of the line with the points(-3, 2) and (2, -8)

)8,2()2,3(

),(),( 2211

and

yxandyx

25

10

)3(2

28

12

12

xx

yym

How did you do? Click the mouse button to see the correct answer.

But please do try it first!