HELLO, GRADE 8 !
Lesson Objectives:
1.
2.
3.
4.
5.
Define equations
Illustrate linear equations in two variables
Illustrate the slope of a linear equation in two variables
Graph linear equations in two variables
Describe the graph of a linear equation
2
LINEAR EQUATIONS
IN TWO VARIABLES
If 𝐴, 𝐵, and 𝐶 are real numbers, and if 𝐴 and 𝐵 are not both
equal to 0, then 𝑨𝒙 + 𝑩𝒚 = 𝑪 is called a linear equation in two
variables. The numbers 𝐴 and 𝐵 are the coefficients of the
variables 𝑥 and 𝑦, respectively, while 𝐶 is the constant.
Examples:
1. 𝑥 + 𝑦 = 5 𝐴 = 1; 𝐵 = 1; 𝐶 = 5
2. 2𝑥 − 5𝑦 = 10 𝐴 = 2; 𝐵 = −5; 𝐶 = 10
5𝑥 + 4𝑦 = 6 𝐴 = 5; 𝐵 = 4; 𝐶 = 6
3. 4𝑦 = 6 − 5𝑥
4
The standard form of a linear equation can be written in slopeintercept form 𝑦 = 𝑚𝑥 + 𝑏 where 𝑚 is the slope and 𝑏 is the y-intercept.
Examples:
1.
𝑥+𝑦 =5
𝑦 = −𝑥 + 5
2.
2𝑥 − 5𝑦 = 10
−5𝑦 = −2𝑥 + 10
3.
4𝑦 = 6 − 5𝑥
4𝑦 = −5𝑥 + 6
−5𝑦 −2𝑥 10
=
+
−5
−5 −5
4𝑦 −5𝑥 6
=
+
4
4
4
5
𝑦=
2
𝑥−2
5
5
3
𝑦=− 𝑥+
4
2
Put me into your standard!
Directions:
Write each of the
following linear
equations in two
variables in
standard form.
6
SLOPE OF A LINE
(Given Two Points, Equation, and Graph)
7
Which road is difficult to drive?
8
Which road will build up speed?
9
Do you think he can send text message to his friends?
10
Is it possible to drive like the illustrations below?
11
Source: backpacker.com
Source: unsplash.com
Source: wikihow.com
12
SLOPE
◉
◉
◉
The slope refers to the steepness of the line.
It is represented by 𝒎, and is defined as the ratio of
the vertical change (rise) between two points to the
horizontal change (run) between the same two points.
In symbol, m =
rise
𝑦 −𝑦
= 2 1.
run
𝑥2 −𝑥1
13
Finding the Slope Given Two Points
Given two points, (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ) the slope of a line can
𝑦 −𝑦
be solved using the formula: 𝑚 = 2 1 .
𝑥2 −𝑥1
Example 1: Find the slope of a line containing points 𝐴(5, 1) and
𝐵(7,9).
𝑥2 𝑦2
𝑥1 𝑦1
Solution:
𝐴 (5, 1)
𝐵 (7, 9)
𝑦2 − 𝑦1 9 − 1 8
𝑚=
=
= =𝟒
𝑥2 − 𝑥1 7 − 5 2
14
Finding the Slope Given Two Points
Given two points, (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ) the slope of a line can
𝑦 −𝑦
be solved using the formula: 𝑚 = 2 1 .
𝑥2 −𝑥1
Example 2: Find the slope of a line containing points 𝐴(5, −1) and
𝐵(7, −9).
𝑥2 𝑦2
𝑥1 𝑦1
Solution:
𝐴 (5, −1)
𝐵 (7, −9)
𝑦2 − 𝑦1 −9 − (−1) −9 + 1 −8
𝑚=
=
=
= −𝟒
=
𝑥2 − 𝑥1
2
7−5
2
15
Finding the Slope Given the Equation
Given the equation of the line written in standard form
(𝐴𝑥 + 𝐵𝑦 = 𝐶), the slope is the ratio of the additive
inverse of the coefficient 𝐴 to coefficient 𝐵. In symbol,
−𝑨
𝒎=
𝑩
16
Finding the Slope Given the Equation
4𝑥 − 𝑦 = 19
Is the equation written in the form 𝐴𝑥 + 𝐵𝑦 = 𝐶?
What are the values of the coefficients A and B?
What is the additive inverse of the coefficient 𝐴?
Get the ratio of the coefficient of the additive inverse 𝐴 to the
coefficient of 𝐵.
Solution:
−𝐴
−4
=
=4
𝐵
−1
17
Finding the Slope Given the Equation
Write the equation in slope intercept form 𝑦 = 𝑚𝑥 + 𝑏.
1. 4𝑥 − 𝑦 = 19
−𝑦 = −4𝑥 + 19
𝑦 = 4𝑥 − 19
Hence, the slope is 4.
2. 20𝑥 + 10𝑦 = 30
10𝑦 = −20𝑥 + 30
10𝑦 −20𝑥 30
=
+
10
10
10
𝑦 = −2𝑥 + 3
Hence, the slope is −2.
RISE OVER RUN
19
Finding the Slope Given the Graph
𝑟𝑖𝑠𝑒
𝑚=
𝑟𝑢𝑛
8
𝑚=
3
The graph of the line
increases/rises
from left to right.
Hence, the slope is
positive
Finding the Slope Given the Graph
𝑟𝑖𝑠𝑒
𝑚=
𝑟𝑢𝑛
3
3
𝑚=
=−
−2
2
The graph of the line
decreases/falls
from left to right.
Hence, the slope is
negative.
Finding the Slope Given the Graph
𝑟𝑖𝑠𝑒
𝑚=
𝑟𝑢𝑛
0
𝑚= =0
4
Finding the Slope Given the Graph
𝑟𝑖𝑠𝑒
𝑚=
𝑟𝑢𝑛
3
𝑚 = = undefined
0
GRAPHING LINEAR EQUATIONS
IN TWO VARIABLES
24
USING ANY TWO POINTS
In graphing a linear equation using this method, you may assign any
two random values of 𝑥, and then solve for the corresponding value of 𝑦.
Example: Graph the linear equation 2𝑥 − 𝑦 = 4.
−𝑦 = −2𝑥 + 4
𝑦 = 2𝑥 − 4
Let us assign any two values of 𝑥 to solve for the value of 𝑦 in the equation 𝑦 =
2𝑥 − 4.
If 𝒙 = 𝟏
𝑦 = 2𝑥 − 4
𝑦 = 2(1) − 4
𝑦 =2−4
𝑦 = −2
(1, −2)
𝐈𝐟 𝒙 = 𝟐
𝑦 = 2𝑥 − 4
𝑦 = 2(2) − 4
𝑦 =4−4
𝑦=0
(2, 0)
USING ANY TWO POINTS
Using the points
(1, −2) 𝑎𝑛𝑑 (2,0), let us graph
the equation 𝑦 = 2𝑥 − 4.
USING x- and y-INTERCEPTS
To find the 𝑥-intercept of a line given its equation, let 𝑦 = 0, then
solve for 𝑥. To find the 𝑦-intercept, let 𝑥 = 0, then solve for 𝑦.
Example: Graph the linear equation 2𝑥 − 𝑦 = 4.
Find the x-intercept:
Let 𝑦 = 0, solve for 𝑥.
𝑦 = 2𝑥 − 4
0 = 2𝑥 − 4
4 = 2𝑥
4
2𝑥
=
2
2
2=𝑥
(2, 0)
−𝑦 = −2𝑥 + 4
𝐅𝐢𝐧𝐝 𝐭𝐡𝐞 𝐲 − 𝐢𝐧𝐭𝐞𝐫𝐜𝐞𝐩𝐭
Let 𝑥 = 0, solve for 𝑦.
𝑦 = 2𝑥 − 4
𝑦 = 2(0) − 4
𝑦 =0−4
𝑦 = −4
(0, −4)
𝑦 = 2𝑥 − 4
USING x- and y-INTERCEPTS
Using the points
(2,0) 𝑎𝑛𝑑 (0, −4), let us graph
the equation 𝑦 = 2𝑥 − 4.
USING SLOPE and its INTERCEPTS
We need to determine the slope and y-intercept of the given
equation. The equation should be written in the form 𝑦 = 𝑚𝑥 + 𝑏.
Example: Graph the linear equation 2𝑥 − 𝑦 = 4.
𝑦 = 2𝑥 − 4
Step 1: Identify the slope and the y intercept.
𝑟𝑖𝑠𝑒
2
𝑚 = 2; 𝑟𝑢𝑛 = 1
y-intercept = −4, hence the ordered pair is (0, −4)
USING SLOPE and its INTERCEPTS
Step 2: Plot first the y-intercept.
Since the y-intercept is −4, we
will plot the ordered pair 0, −4 .
Step 3: From the y-intercept
0, −4 , use the slope to plot the
next point. Since the slope is
positive 2, move 2 units upward
from the y-intercept and 1 unit to
the right, and then mark that
point.
Describing the Graph of a Linear Equation
Trend of the Graph
The value of the slope 𝑚 tells the trend of the
graph of a linear equation.
31
If 𝒎 is negative, then the graph is decreasing
from left to right.
If 𝒎 is positive, then the graph is increasing
from left to right.
32
If 𝒎 is undefined, then the graph is a vertical
line.
If 𝒎 is zero, then the graph is a horizontal line.
33
I LOVE MATH !
I LOVE MATH !
I LOVE MATH !
34