NORTHERN PHILIPPINES ACADEMY
Centro Norte Gattaran, Cagayan
3508
SELF - LEARNING MODULES IN MATHEMATICS 8
Teacher: MARICEL A. COLUMNA
Face book Account : Maricel Agag Columna
CP #: 09653107067
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QUARTER 2: MODULE 1and 2
Reminders:
a. Read the instructions carefully before starting anything.
b. Complete all the activities.
c. Answers on these modules shall be placed in separate sheet of paper and don’t forget to write your name in your
answer sheet.
d. For questions and queries, you may contact me through my face book account and contact number indicated above.
Please follow the schedules of your subject areas given to you.
e. Allow time for relaxation and recreation when you are mentally tired.
I.
Introduction: Have you ever wondered how businessmen nowadays make decisions? What guides businessmen
in making decisions? What are the tools that they use? How do they know when to increase or decrease production? In
this module, you will discover how important it is to utilize essential mathematical skills to be able to understand these
questions that arise in various real – life situations that we encounter every day and use these skills wisely to be able to
come up with the desired output.
II.
Lesson Objectives/MELCs:
a. Graphs a linear equation given any two points; x – and y – intercepts ; slope and a point on the line
b. Describes the graph of a linear equation in terms of its intercepts and slope
c. Finds the equation of a line given two points, the slope and a point on the line, and the slope and its y –
intercepts
d. Solves word problems that involve linear equations in two variables
e. Illustrates and graphs a system of linear equations in two variables
f. Categorizes when a given system of linear equations have graphs that are parallel, intersecting or
coinciding
g. Solves a system of linear equations by graphing, by elimination method and by substitution
h. Differentiates a linear equation in two variables from a linear inequality in two variables
i. Illustrates and graph a linear inequality in two variables
III.
Lesson Proper
Module 1 – Week 1
Lesson 1: Linear Equations in Two Variables and its Graphs
Pre – Activity: Determine the given statement if it’s correct or not then write T or F before the item number.
1.
2.
3.
4.
5.
A linear equation can be easily graph by assigning real values for the independent variable x.
Graphing linear equations is similar to plotting points on a number line.
The slope of a line is defined as the ratio of its rise to its run.
A negative rise means the direction is upward.
A linear equation can be also graphed using two given points, or the slope of the line and a point on the line.
Open your textbook on pages 77 – 106. Study and examine the following examples.
Remember:

You can graph a linear equation by assigning values of x to find y, refer on page 77 – 79. The values as shown in
the table will be used in graphing the equation. Plot the points on the plane and connect all the points by
drawing a line.



In graphing an equation that passes through two points, all that needs to be done is to plot each of the points on
the coordinate plane and draw a line passing through both points.
In graphing equation of the line with intercepts ( x – and y – intercepts), refer on page 85 – 86. Thus, to graph
the equation, simply plot each of the points and draw a line passing through them.
In graphing a linear equation given the slope and a point, plot the given point and use the given slope to find the
second point, refer on examples on pages 84 – 85. Analyze the given solution and graph illustrated in your
textbook.
Activity 1: Use separate paper/graphing paper.
Graph the linear equation by using the 3 values for x shown in the table. Complete the table to show the values of
𝑥, 𝑦 𝑎𝑛𝑑 (𝑥, 𝑦). The first one is done for you.
𝒙
2
1
0
-1
-2
Given: 𝒚 = 𝟑𝒙 – 𝟐
Solution: x = 2
y = 3x – 2
y = 3(2) – 2
=6–2
y=4
𝒚
4
(𝒙, 𝒚)
(2, 4)
Lesson 2: Equations of a Line
Pre – Activity: Match column A into column B. Write the letter of your answer on the blank.
______
Column A
𝑦 −𝑦
1. 𝑦 − 𝑦1 = 𝑥2 −𝑥1 (𝑥 − 𝑥1 )
2
Column B
a. General/Standard Form
1
b. Intercept Form
2. 𝑦 − 𝑦1 = 𝑚(𝑥2 − 𝑥1 )
c. Two – Point Form
3. 𝐴𝑥 + 𝐵𝑦 = 𝐶
d. Point – Slope Form
4. 𝑦 = 𝑚𝑥 + 𝑏
𝑥
𝑦
e. Slope – Intercept Form
5. 𝑎 + 𝑏 = 1
Open your textbook on pages 89 – 95. Read and analyze the different ways to determine the equation of a line.
______
______
______
______
𝑦 −𝑦

Given Two Points (Two – Point Form): 𝑦 − 𝑦1 = 𝑥2 −𝑥1 (𝑥 − 𝑥1 ) , refer on page 91.


Given a Point and the Slope: 𝑦 − 𝑦1 = 𝑚(𝑥2 − 𝑥1 ) , refer on page 93.
Given the slope and the y – Intercept: 𝑦 = 𝑚𝑥 + 𝑏, refer on pages 94, examples 3.22 and 3.23.

Given Two Intercepts ( 𝑥 𝑎𝑛𝑑 𝑦): 𝑎 + 𝑏 = 1 – refer on pages 94 – 95.

The equation of a line can be determined given any of the following: two points on the line, the slope of the line
and a point on it, the slope of the line and its y – intercept, or its x – and y – intercepts.
2
𝑥
1
𝑦
Activity 2: Determine the equation of the line. Use separate sheet of paper.
1. Determine the equation of the line that passes through (8, 4) and (-2, 0). (use Two – Point Form)
1
2
2. Find the equation of the line whose y – intercept is 14 and whose slope is . (use slope and y – intercept form)
Module 1 – Week 2
Lesson 3: Applications of Linear Equations in Two Variables
Solving word problems in two variables entails proper interpretation of the given information, along with identifying the
part of the equation that represents a particular piece of information. How to solve problems that involve linear
equations in two variables? The first step is to determine the information given and the quantities to be found. From
there, we proceed by relating the given information to the characteristics of a linear equation in two variables.
Open your textbook on pages 104 – 106. Each example shows the step or solution in solving a given linear equation.
Study carefully the examples presented so that you may fully understand the lesson.
Note: To solve mathematical problems orderly, you have to follow these steps:
R – representation (assigning variables for the unknown values, “x and y”
E – equation (translating the given problem mathematically)
S – solution (computational part)
A – answer (statement form of the answer to the given problem)
Activity 3: On page 106, choose one (1) problem from item 1 – 4 and solve it by using the RESA steps.
Module 2 – Week 3
Lesson 4: Kinds of Systems of Linear Equations in Two Variables
Pre – Activity: Identify whether the statement is true or false. Write your answer before the item number.
1. An independent and consistent system has only one solution.
2. A system of linear equations is a group of two or more linear equations that share the same variables.
3. A system of linear equations whose graphs are coincident lines is called a consistent and dependent system.
4. There are three methods of solving systems of linear equations namely, a graphing, substitution and elimination.
5. Systems of linear equations are classified according to the number of solutions.
Open your textbook on pages 128 – 132. Study the definition and types of systems of linear equations and examine the
following graphs of each system.
 A system of linear equations in two variables is a collection of linear equations that share the same variables.
 The table below summarizes the results of the different types of systems of linear equations in two variables.
Number of
Type of System
Slope
Graph/relationship of lines
𝒚 – 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕
Solutions
Inconsistent
No solution or
Same or =
Different or ≠
Parallel lines
zero
Consistent and
Different (≠) or
One solution
Different or ≠
Intersecting lines
Independent
same(=)
Consistent and
Infinite or many
Same
Same
Coincident/coinciding lines
Dependent
solutions
Activity 4A: Without graphing, determine the kind of system, the number of solutions, and the type of graph formed by
the given systems below.
2𝑥 − 7𝑦 = −3
4𝑥 − 8𝑦 = 1
1. {
2. {
6𝑥 = 21𝑦 − 3
8𝑥 − 4𝑦 = 1
 There are three methods in solving linear equations. Study the given examples and solution on the following
methods.
a. Graphical/Graphing Method – refer on pages 134 – 137
b. Substitution Method – refer on pages 139 - 142
c. Elimination Method – refer on pages 143 – 144
 Additional readings are on pages 98 – 101. Study this topic, Parallel and Perpendicular lines.
Activity 4B: Solve the following systems using the given method. Show your solution in your answer sheet.
1. {
𝑥−𝑦 =7
by substitution
𝑦 = 3𝑦 + 15
2. {
𝑥 + 𝑦 = 11
by elimination
−𝑥 + 𝑦 = −1
Module 2 – Week 4
Lesson 5: Linear Inequalities in Two Variables: Its Graph and Solutions
Open your textbook on pages 107 – 112. Examine the examples and solutions on how to graph linear inequality.
Note:
 A linear equation is an algebraic equation whose graph is a straight line and whose independent variable has a
degree of 1 and it uses an equality symbol (=).
 A linear inequality on the other hand is a mathematical statement involving linear expression that uses any of
the inequality symbols such as <, >, ≤, ≥ instead of the equality symbol.



The solution set of a linear inequality is a half – plane. It may or may not include the boundary line, which is the
line that divides the plane into two halves.
A strict inequality is an inequality that uses either of the symbols< 𝑜𝑟 >. Its solution set does not include the
boundary line of its graph.
A nonstrict inequality is an inequality that uses either of the symbols ≤ 𝒐𝒓 ≥. Its solution set includes the
boundary line of its graph.
Activity 5: Answer Progress Check on pages 112 – 113 A only, item 1 – 10.
IV.
Assessment
A. Read carefully and analyze each statement. Write the letter of your answer before the number.
1. The statement “the solution set of an equation includes any point on the line that is determined by that
equation” is __________.
a. sometimes true
b. sometimes false
c. always true d. always false
2. Which of the following does not determine a unique linear equation?
a. x and y intercepts
b. two points on the line
c. m and a point on the line
d. m and the x intercept
c. 𝑦 − 𝑦1 = 𝑚(𝑥2 − 𝑥1 )
d. 𝐴𝑥 + 𝐵𝑦 = 𝐶
3. The point – slope form of a linear equation is ______.
a. 𝑦 = 𝑚𝑥 + 𝑏
𝑦 −𝑦
b. 𝑦 − 𝑦1 = 𝑥2 −𝑥1 (𝑥 − 𝑥1 )
2
1
4. The slope of the line with equation of 4x – 3y = 12 is ___________.
a. 3
4
b. 4
c. 3
3
d. 4
5. The slope of a line that passes through the points (1, 7) and (7, 1) is _______.
a. 1
b. – 6
c. 6
d. – 1
B. Answer the questions briefly.
1. How will you describe the relationships of the slopes and the y – intercepts of parallel and perpendicular lines?
2. When determining the kind of system of two linear equations, is it still necessary to examine the relationship of
their y – intercepts if their slopes are already unequal? Why or why not?
Congratulations! You have made it! Be ready on your next modules…………….
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Textbook Reference: K – 12 ADVANCE with MATH 8 (REX Book Store)
Authors :
Edmond S. Bunag
Ma. Edilyn Dimapilis – Chiao
Francis Joseph H. Campeña, Ph.D.
Websites for Online Math Links, see page 124
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Prepared by:
MARICEL A. COLUMNA
Subject Teacher
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NOTE:
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Summative Test/Monthly Test for this month of November please refers on your textbook.
Assessment of Learning (Write the letter of your answer only)
 Pages 119 – 124 item 1 – 20 and
 Pages 166 – 171 item 1 – 20
Write your answers in your answer sheet along with the modules.
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