Graphing Linear Equations. X and Y Coordinates A coordinate or ordered-pair notation is always...
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![Page 1: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/1.jpg)
Graphing Linear Equations
![Page 2: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/2.jpg)
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.
![Page 3: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/3.jpg)
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
![Page 4: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/4.jpg)
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!
![Page 5: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/5.jpg)
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.
![Page 6: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/6.jpg)
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)
![Page 7: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/7.jpg)
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.
![Page 8: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/8.jpg)
The Rectangular Coordinate System
Graphing
![Page 9: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/9.jpg)
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.
![Page 10: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/10.jpg)
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.
![Page 11: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/11.jpg)
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.
![Page 12: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/12.jpg)
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)
![Page 13: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/13.jpg)
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.
![Page 14: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/14.jpg)
Vertical and Horizontal Lines
Vertical lines have an equation of x=c.
Horizontal lines have an equation of y=c.
y = c
x = c
![Page 15: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/15.jpg)
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.
![Page 16: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/16.jpg)
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
![Page 17: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/17.jpg)
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.
![Page 18: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/18.jpg)
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.
![Page 19: 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.](https://reader038.fdocuments.us/reader038/viewer/2022103015/551c1bd7550346ad4f8b58eb/html5/thumbnails/19.jpg)
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!