8

NOT

Mathematics
Quarter 1 - Module 9
Graphing Linear Equations
in Two Variables
Mathematics - Grade 8
Alternative Delivery Mode
Quarter 1 – Module 9: Graphing Linear Equations in Two Variables
First Edition, 2020

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 8

Mathematics
Quarter 1 - Module 9
Graphing Linear Equations
in Two Variables
This page is intentionally blank
 Table of Contents

What This Learning Package is About…………………………………………………………...i

What I Need to Know………………………………………………………………………………...i
How to Learn from this Learning Package……………………………………………………...i
Icons of this Learning Package…………………………………………………………………...ii

What I Know…………………………………………………………………………………………..iii

Lesson 1:
Graphing Linear Equations in Two Variables …………………………...1
What I Need to Know…………………………………………………………….1

What’s New: Activity 1 Describe Me …………………………………………1

What Is It: Using Two Points……..……………………………………………2

What’s More: Activity 2 Sketch Me (Part 1) ……………………………..3

What Is It: Using x- and y-intercept … …………………………………..4

What’s More: Activity 3 Sketch Me (Part 2) …………………………….6

What Is It: Using Slope and One Point ….……………………………...7

What’s More: Activity 4 Sketch Me (Part 3) …………………………....8

What Is It: Using Slope and One Point ….……………………………...9

What’s More: Activity 5 Sketch Me (Part 3) ……………………….…..11

What I Have Learned: Activity 6 Generalization …………………….…. 12

What I Can Do: Activity 7 Light Me Up …………………………………… 13

Summary……………………………………………………………………………………14
Assessment: (Post-Test)………………………………………………………………...14
Key to Answers……………………………………………………………………………………17
References………………………………………………………………………………………….20
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What This Module is About
This module discusses the four (4) methods on how to graph linear equations
in two variables. Each method will be discussed with a given example.

You are expected at the end of this module to sketch the graphs of the linear
equations in two variables.

What I Need to Know

At the end of this module, you should be able to:

1. Graph a linear equation given

(a) any two points;
(b) the x – and y – intercepts;
(c) the slope and y – intercept; and
(d) the slope and a point on the line. (M8AL-If-2)

How to Learn from this Module

To achieve the objectives cited above, you are to do the following:
• Take your time reading the lessons carefully.
• Follow the directions and/or instructions in the activities and exercises diligently.
• Answer all the given tests and exercises.

i
Icons of this Module
What I Need to This part contains learning objectives that
Know are set for you to learn as you go along the
module.

What I know This is an assessment as to your level of

knowledge to the subject matter at hand,
meant specifically to gauge prior related
knowledge
What’s In This part connects previous lesson with that
of the current one.

What’s New An introduction of the new lesson through

various activities, before it will be presented
to you

What is It These are discussions of the activities as a

way to deepen your discovery and under-
standing of the concept.

What’s More These are follow-up activities that are in-

tended for you to practice further in order to
master the competencies.

What I Have Activities designed to process what you

Learned have learned from the lesson

What I can do These are tasks that are designed to show-

case your skills and knowledge gained, and
applied into real-life concerns and situations.

ii
 What I Know

Pre-Assessment
Directions: Read and understand each question carefully and write the letter of your
answer in your answer sheet.
1. The following are the methods on how to graph a line, except
a. using given intercepts c. using slope and a point
b. using given points d. using random ordered pairs
2. Which of the following is a linear equation in two variables?
a. 3x2 = y c. 8x + 9y = 3z
b. y = 7x – 7 d. x + y
3. The graph of a linear equation is a ________.
a. parabola c. line
b. curve d. broken line
4. The graph of the points (3, 5) and (7, 1) is ______.
a. 7

6
c. 7

5
5
4
4
3
3
2
2
1 1 2 3 4 5 6 7 8
1 1 2 3 4 5 6 7 8 -5 -4 -3 -2 -1 -1
-5 -4 -3 -2 -1 -1
-2
-2
-3
-3
-4
-4
-5
-5

7 6

b. 6

4
d. 5

3 2

2 1 1 2 3 4 5 6 7 8
-5 -4 -3 -2 -1 -1
1 1 2 3 4 5 6 7 8
-5 -4 -3 -2 -1 -1 -2

-2 -3

-3 -4

-4 -5

-5

5. A linear equation in two variables written in the form y = mx + b is in ________.

a. standard form c. point-slope
b. Slope-intercept form d. intercepts form
6. The graph of the line x = 4 is a line _________.
a. parallel to x-axis at a distance of 4 units from the origin.
b. making an intercept 4 on the x-axis
c. parallel to y-axis at a distance of 4 units from the origin.
d. making an intercept 4 on both the axis.
7. The ______ of a linear equation is Ax + By = C, where A and B are not both zero.
a. run c. standard form
b. slope d. slope-intercept form
8. The x- and y-intercepts of the linear equation 4x – 3y = 12 is _______.
a. (3,0), (0,-4) c. (3,0),(0,4)
b. (-3,0), (0,-4) d. (-3,0),(0,-4)
9. Given two points (x1, y1) and (x2, y2), what is the formula for the slope?
𝑥 −𝑥 𝑥 −𝑦 𝑦 −𝑦 𝑦
a. 𝑚 = 1 2 b. 𝑚 = 1 1 c. 𝑚 = 1 2 d. 𝑚 =
𝑦1 −𝑦2 𝑥2 −𝑦2 𝑥1 −𝑥2 𝑥
10. If x = - 2, what will be the value of y in the equation y = -3x – 8?
a. 2 b. – 2 c. – 14 d. 14

iii
 For numbers 11 to 15, identify the linear equation. Refer your answers on the
given graph.

11. -10
-9
a. y = -7x -7
-8
3
-7
-6 b. y = − 𝑥 − 3
-5 7
-4
7
-3
-2 c. 𝑦 = − 𝑥 − 7
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1 3
-1
-2
1 2 3 4 5 6 7 8 9
d. y = -x – 3
-3
-4
-5
-6
-7
-8
-9
-10

12. a. y = x + 6
b. y = -x -6
-10
-9
-8
-7
-6
-5
c. y = -x + 6
d. y = x – 6
-4
-3
-2
-1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7 8 9 10
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10

13. -10
a. y = x + 3
b. y = x – 3
-9
-8
-7

c. y = -x – 3
-6
-5
-4
-3

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-2
-1 d. y = -x + 3
1 2 3 4 5 6 7 8 9 10
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10

10

14. 9
8
7
a. y = 7x + 7
6
5
4
b. y = x + 2
3
2
2
1 c. 𝑦 = − 𝑥 + 2
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7 8 9 10
7
-1
7
-2
-3 d. 𝑦 = − 𝑥 + 7
-4 2
-5
-6
-7
-8
-9

15. 10 a. y = -4
9
8 b. x = -4
7
6 c. y = 4
5
4
3
d. x = 4
2
1
-8 -7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7
-1
-2
-3
-4
-5
-6
-7
-8
-9

iv
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 Graphing Linear Equations
Lesson in Two Variables
1
What I Need to Know

In the previous module, you have learned that there are many ways in
representing functions. One of which is through an equation. A linear equation in two
variables can be written in its standard form Ax + By = C or slope-intercept form
y = mx + b. It can also be graphed using the following methods:
a. using any two points;
b. using the x and y intercepts;
c. using the slope and y - intercept; and
d. using the slope and a given point.

What’s New

5
Activity 1 Describe Me! 4

Description: This activity will enable you to describe the 2

graph of a linear equation in two variables in terms of its -5 -4 -3 -2

1 1
-1 -1
2 3 4

intercepts, slope, and points. -2

-3

Directions: Given the graph at the right, find the following: -4

-5

Graph created through MSExcel

1. x – intercept 4. run
2. y – intercept 5. slope
3. rise 6. trend

1
 What Is It

Activity 1 lets you describe the graph of a linear equation in two variables. Its
graph is a line, and a line is created using two points. This leads to the first method of
graphing a linear equation in two variables which is using two points.

Using Two Points

Steps in graphing linear equations in two variables using two points.

1. Assign two values for x. (x is any real number)
2. Solve the values of y by substituting the assigned values of x in the equation.
3. Plot the two points (solutions of the equation) in the rectangular coordinate
system.
4. Connect the two points to form a straight line.

Illustrative Example
Graph the linear equation in two variables y = x + 2 using two points.

Step Solution
1 Let x = 1 and x = - 2.

2 If x = 1, then If x = - 2, then
y=x+2 y=x+2
y=1+2 y=-2+2
y=3 y= 0
Hence, the first point Hence, the second point
(solution) is (1, 3). (solution) is (- 2, 0).

2
 4

Graph created from https://www.desmos.com/calculator.

What’s More

Activity 2 Sketch Me (Part 1)

Given two values for x, graph the linear equations in two variables in a graphing
paper. The first item serves as an example, answer numbers 2 – 4.

1. y = - 3x + 4
x=2 x = -1

y = - 3x + 4 y = - 3x + 4
y = - 3(2) + 4 y = - 3(-1) + 4
y=-6+4 y=3+4
y=-2 y=7

(2, -2) (- 1, 7)
Graph created from https://www.desmos.com/calculator.

2. x – y = 5
x=3 x=-2

3
3. 3x + 4y = 0
x=4 x=-3

4. 4x – 2y = 6
x=4 x=-3

What Is It

The x – intercept is the abscissa of the coordinates of the point (a, 0) where
the graph intersects the x – axis while the y – intercept is the ordinate of the
coordinates of the point (0, b) where the graph intersects the y – axis. The next method
of graphing makes use of the intercepts.

Using x – intercept and y – intercept

Steps in graphing linear equations in two variables using x – intercept and

y - intercept.
1. Solve for y – intercept by letting x = 0 to have a point (0, b).
2. Solve for x – intercept by letting y = 0 to have a point (a, 0).
3. Plot the two points in the rectangular coordinate system.
4. Connect the two points to form a straight line.

4
Illustrative Example

Graph the linear equation in two variables 2𝑥 – 𝑦 = 4 using x - intercept and

y - intercept.

Step Solution
1 Let x = 0,
2x – y = 4
2(0) – y = 4
–y=4
y = - 4. Hence, (0, - 4).
2 Let y = 0,
2x – y = 4
2x – 0 = 4
2x = 4
x=2 Hence, (2, 0).
3

Graph created from https://www.desmos.com/calculator.

5
 What’s More

Activity 3 Sketch Me (Part 2)

Graph the linear equations in two variables using the x – intercept and y –
intercept. The first item serves as an example, answer numbers 2 – 4.

1. x + y = 8
x – intercept: a = 8 y – intercept: b = 8

x+y=8 x+y=8
x+0=8 0+y=8
x=8 y=8

(8, 0) (0, 8)

3
2. y = x
4
x – intercept: a = y – intercept: b =

3. 5x = 20 – 2y
x – intercept: a = y – intercept: b =

4. 3x + 6y + 9 = 0
x – intercept: a = y – intercept: b =

6
 What Is It
You have learned that the standard form of the linear equation of two variables
𝑨𝒙 + 𝑩𝒚 = 𝑪 can be rewritten in its slope-intercept form which is 𝒚 = 𝒎𝒙 + 𝒃. The third
method of graphing uses the slope and y – intercept.

Using the Slope and y - intercept

Steps in graphing linear equations in two variables using the slope and
y - intercept.
1. Write the equation in the slope-intercept form 𝒚 = 𝒎𝒙 + 𝒃.
2. Identify the slope m and y – intercept b.
3. Plot the point (0, b) in the rectangular coordinate system.
rise
4. From the point (0, b), plot another point using the slope 𝒎= .
run
A third point is use for checking.
5. Connect the points to form a straight line.

Illustrative Example

Graph the linear equation in two variables 4𝑥 + 5𝑦 = 20 using the slope and
y - intercept.

Step Solution
1 4x + 5y = 20
5y = - 4x + 20
4
y = − x+4
5

2 4
𝑚 =− and b = 4
5
3

or

7
 5

Graph created from https://www.desmos.com/calculator.

What’s More

Activity 4 Sketch Me (Part 3)

For each linear equation in two variables, find the slope, y – intercept and then
graph the line. The first item serves as an example, answer numbers 2 – 4.

1. 3x + 2y = 18
slope: m = -
3 y – intercept: b = 9
2

3x + 2y = 18
2y = - 3x + 18
3
y=- x+9
2

2. 2𝑥 − 5𝑦 = 15
slope: m = y – intercept: b =

8
3. x + 4y = - 4
slope: m = y – intercept: b =

4. y = 7x + 9
slope: m = y – intercept: b =

What Is It
The fourth method of graphing makes use the slope and a point on the line.
The point is a solution of the linear equation in two variables.

Using the Slope and a point

Steps in graphing linear equations in two variables using the slope and
a point on a line.
1. Write the equation in the slope-intercept form 𝒚 = 𝒎𝒙 + 𝒃 and identify the
rise
slope 𝒎= .
run
2. To find a point, assign a value for x and substitute it in the equation to solve
for y. You will have point (x, y).
3. Plot point (x, y) in the rectangular coordinate plane.
rise
4. From point (x, y), plot another point using the slope 𝒎= .
run
A third point is use for checking.
5. Connect the points to form a straight line.

9
Illustrative Example
Graph the linear equation in two variables -8x + 6y = - 12 using the slope and
a point.

Step Solution
1 -8x + 6y = - 12
6y = 8x - 12
4
y= x –2
3

4
slope: 𝑚 =
3

2 Let x = 3, or you can use the equation

-8x + 6y = - 12 in its slope-intercept form
-8(3) + 6y = - 12 4
y= x –2
-24 + 6y = - 12 3
6y = 12 4
y = (3) – 2
y =2 3
12
y= –2
3

y=4–2
y=2
(3, 2) (3, 2)
3

10
 5

Graph created from https://www.desmos.com/calculator.

What’s More

Activity 5 Sketch Me (Part 4)

For each linear equation in two variables, find the slope, a point and then graph.
The first item serves as an example, answer numbers 2 – 4.

1. x - 3y = 15
x - 3y = 15 Let x = 6
- 3y = - x + 15 1
y= x–5
1 3
y= x–5 1
3 y= (6) –5
3
6
1 y= –5
slope: m = 3
3 y=2–5
y=-3
(6, -3)

2. y = 5x + 2
Slope: m = Let x = 1

11
 3. 2x – y = 2
slope: m = Let x = 2

4. 2x + 3y = -3
slope: m = Let x = 3

What I Have Learned

Activity 6 Generalization

Fill in the diagram below the steps in graphing linear equations in two variables
using the four (4) different methods.

Using Two Using the

Points Slope &
y-intercept

Using the
Using Slope &
a point
Intercepts

12
 What I Can Do

Activity 7 Light Me Up

Description: This activity will enable you to solve real-life problems involving linear
functions.
Directions: Consider the situation below and answer the questions that follow.

Zeke lights a 9 inches long candle. He records the length of the candle,
y represents the length while x represents the hours.

Number of Hours Candle Burns

0 1 2 3 4
(x hours)

Height of the Candle

9 7 5 3 1
(y inches)

1. What is the dependent variable? Explain your answer.

___________________________________________
2. What is the independent variable? Explain your answer.
____________________________________________
3. Based on the completed table, would the relation represent a line?
____________________________________________________
4. What is the y-intercept? Explain your answer.
____________________________________
5. What is the slope? Explain your answer.
________________________________
6. Write the linear equation.
____________________

13
Summary
A linear equation in two variables can be graphed using:

✓ two points on the line

- any given ordered pairs which are solutions of the linear
equation
✓ the x- and y-intercept.
- x- intercept is the abscissa of the coordinates while
y-intercept is the ordinate of the coordinates of the points
where the line passes through the x and y axes respectively
✓ slope and y – intercept
- first, plot point (o, b) where b is the y-intercept and then use
the slope to find another point
✓ a point and the slope
- the graph can be done by plotting first a point which is a
solution of the linear equation and then using the slope from
the point to find another point

Assessment: (Post-Test)

Directions: Read and understand each question carefully and write the letter of your
answer in your answer sheet.

1. A line represents the graph of __________.

a. quadratic equation c. monomial
b. polynomials d. linear equation

2. The __________ of the line is the value of x when y = 0.

a. y-intercept c. slope
b. x-intercept d. none of the above

3. In graphing linear equations in two variables using slope and y-intercept form,
if m > 0, the graph is a line that rises from _______ to _______.
a. right, left c. left, right
b. Top, bottom d. none of the above

4. In graphing linear equations in two variables using slope and y-intercept form,
if m < 0, the graph is a line that rises from _______ to _______.
a. top, bottom c. right, left
b. left, right d. none of the above

14
 5. State the four (4) methods on how to graph linear equations in two variables.

For numbers 6 to 10, identify the linear equation. Refer your answers on the
given graph.
3
6. a. 𝑦 = 𝑥 + 3
2
b. 𝑦 = 𝑥 + 3
c. 𝑦 = −2𝑥 – 2
2
d. 𝑦 = 𝑥 –2
3

7. a. y = -4
b. x = -4
c. y = 4
d. x = 4

8. a. y = 7x + 7
b. y = x + 2
2
c. 𝑦 = − 𝑥 + 2
7
7
d. 𝑦 = − 𝑥 + 7
2

9. a. y = x -3
b. y = -x -3
c. 𝑦 = 𝑥 + 3
d. 𝑦 = −𝑥 + 3

10. a. y = x + 6
b. y = -x – 6
c. y = -x + 6
d. y = x – 6

15
For numbers 11 to 15. Tell whether the statement is TRUE or FALSE.
11. The graph of 0x + 1y = 8 is a horizontal line.
12. The graph of the equation x = -6 is a vertical line.
13. The graph of a linear equation in two variables is a line.
14. Given one point, a graph of a linear equation can be sketched.
15. In the slope-intercept form of an equation, b represents the y-intercept.

16
 17
Pre-Assessment:
1. d 6. c 11. a
2. b 7. c 12. b
3. c 8. a 13. b
4. a 9. c 14. c
5. b 10. b 15. b
Lesson 1
Activity 1 Describe Me!!!
1. x-intercept: -3 4. run: 3
2
2. y-intercept: 2 5. slope:
3
3. rise: 2 6. trend: increasing
Activity 2 Sketch Me (Part 1)
2. (3, -2) (-2, -7) 3. (4, -3) (-3, 9/4)
4. (4, 5) (-3, -9)
Key to Answers
 18
Activity 3 Sketch Me (Part 2)
2. (0,0) 3. (0,10) (4, 0) 4. (0,-3/2) (-3, 0)
Activity 4 Sketch Me (Part 3)
2. m = 2/5 b = -3 3. m = -1/4 b = -1 4. m = 7 b = 9
Activity 5 Sketch Me (Part 4)
2
2. m = 5; (1,7) 3. m = 5; (2,2) 4. m = − ; (3,-3)
3
 19
Activity 6
Steps in graphing linear equations in two Steps in graphing linear equations in two
variables using x – intercept and variables using the slope and
y - intercept. y - intercept.
1. Solve for y – intercept by letting x = 0 to have 1. Write the equation in the slope-intercept form
a point (0, b). 𝒚 = 𝒎𝒙 + 𝒃.
2. Solve for x – intercept by letting y = 0 to have 2. Identify the slope m and y – intercept b.
a point (a, 0). 3. Plot the point (0, b) in the rectangular
3. Plot the two points in the rectangular coordinate system.
coordinate system. 4. From the point (0, b), plot another point using
4. Connect the two points to form a straight line. rise
the slope 𝒎= .
run
A third point is use for checking.
5. Connect the points to form a straight line.
Steps in graphing linear equations in two Steps in graphing linear equations in two
variables using x – intercept and variables using the slope and
y - intercept. a point on a line.
1. Solve for y – intercept by letting x = 0 to have 1. Write the equation in the slope-intercept form
a point (0, b). rise
𝒚 = 𝒎𝒙 + 𝒃 and identify the slope 𝒎= .
run
2. Solve for x – intercept by letting y = 0 to have
2. To find a point, assign a value for x and
a point (a, 0).
substitute it in the equation to solve for y. You
3. Plot the two points in the rectangular
will have point (x, y).
coordinate system.
3. Plot point (x, y) in the rectangular coordinate
4. Connect the two points to form a straight line.
plane.
4. From point (x, y), plot another point using the
rise
slope 𝒎= .
run
A third point is use for checking.
5. Connect the points to form a straight line.
Activity 7
1. The dependent variable is the height of the candle because it depends on the
number of hours the candle is burning.
2. The independent variable is the number of hours the candle is burning because
it controls the height of the burning candle.
3. Yes. It represents a line.
4. The y-intercept of the line is 9.
5. The slope is -2.
6. The equation is y = 9x – 2.
Post-Test
1. d 6. c 11. False
2. b 7. c 12. True
3. c 8. b 13. True
4. c 9. d 14. False
5. Using two points, 10. d 15. True
x- and y-intercept,
slope and one point,
and slope and y-intercept.
References:
Abuzo, Emmanuel, Merden Bryant, Jem Boy Cabrella, Belen Caldez, Melvin
Callanta, Anastacia Proserfina Castro, Alicia Halabaso, Sonia Javier, Roger
Nocom, and Concepcion Ternida. Mathematics Learner's Module 8. 1st ed.
Reprint, Department of Education, 2013.

Firmalino, Sandra Bernadette F, May Maricel B De Gracia, Elizabeth P Jimenez, and

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