NAGA VIEW ADVENTIST COLLEGE, INC.
Panicuason, Naga City
LEARNING MODULE
Mathematics G8 | Q2 | Module 1
Systems of Linear Equations
in Two Variables
Name:
content STANDARD:
The learner demonstrates understanding of key concepts of systems of linear equations.
PERFORMANCE STANDARD:
The learner is able to formulate real-life problems involving systems of linear equations in two variables
and solve these problems accurately using a variety of strategies.
General Instruction:
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Activities in blue box are for group discussion, group sharing, and/or selfassessment. No need to submit. Answers are found in the Answer Key.
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Activities in green box need to be submitted.
• Have fun while learning! 🤩 May God bless your studies. 😇
This module is divided into four (4) sections:
• Explore – The lesson starts with activity where you are given open-ended question/s and an
opportunity to share your ideas.
• Firm-up – The part where you learn new ideas and concepts about the lesson.
• Deepen – Your goal in this section is to look at some real-life situations where we can apply
the concepts that you have learned.
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Transfer – Your goal in this section is to apply your learning to real-life situations. You will be
given a practical task which will demonstrate your understanding.
vocabulary list:
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Coinciding lines – overlapping lines.
Consistent and Dependent System – a system of linear equation having infinitely many solutions. The
slopes of the lines defined by the equations are equal and their y-intercepts are also equal and their graphs
coincide.
Consistent and Independent System – a system of linear equations having exactly one solution. The
slopes of the lines defined by the equations are not equal, their y-intercepts could be equal or unequal, and
their graphs intersect at exactly one point.
Inconsistent System – a system of linear equation having no solution. The slopes of the lines defined by the
equations are equal or both lines have no lines, their y-intercepts are not equal, and their graphs are parallel.
Intersecting lines – lines that meet at exactly one point.
Parallel Lines – coplanar lines that do not intersect.
Solution – the coordinates of all points of intersection of the graphs of the equations in the system whose
coordinates must satisfy all equations in the system.
System of Linear Equations – also called as simultaneous linear equations. A set or collection of linear
equations, all of which must be satisfied.
x-intercept – the value of x when the value of y is zero of a linear equation. The x-coordinate of the point
when the line intersects the x-axis.
y-intercept – the value of y when the value of x is zero of a linear equation. The y-coordinate of the point
when the line intersects the y-axis.
To familiarize yourself with the terms above, go to this link and have fun.
https://wordwall.net/play/23643/229/341
explore
We often see this kind of puzzle in our Facebook feed. Let's try this out. Can you solve this
puzzle? What is the value of each emoji "Sunglasses Smile" and "Tongue Out Smile"?
Activity 1.
EMOJI LOGIC PUZZLE
Directions: Answer the puzzle below and the questions that follow.
Questions:
a) What is the value of the emoji “Sunglasses Smile”? _________________
b) What is the value of the emoji “Tongue Out Smile”? _________________
c) How were you able to find their values?
How was the activity? Did you have fun solving the puzzle? This kind of puzzle or problem can be
answered best by Systems of Linear Equations in Two Variables.
firm-up
This section will help you learn new concepts about the Systems of Linear Equations in Two
Varibles.
A system of linear equations in two variables consists of two or more linear equations made up
of two variables such that all equations in the system are considered simultaneously. To find the unique
solution to a system of linear equations, we must find a numerical value for each variable in the system that
will satisfy all equations in the system at the same time. Some linear systems may not have a solution and
others may have an infinite number of solutions. In order for a linear system to have a unique solution,
there must be at least as many equations as there are variables. Even so, this does not guarantee a unique
solution.
The solution to a system of linear equations in two variables is any ordered pair that satisfies each
equation independently and may be interpreted graphically as the intersection between the two lines.
Let us revisit Activity 1: Emoji Logic Puzzle and convert the emojis to variables.
Let x = Sunglasses Smile emoji and y = Tongue Out emoji
𝒙+𝒚=𝟓
3𝒙 + 𝒚 = 9
x+y=5
. The brace denotes that the two equations can
3x + y = 9
be solved simultaneously. Let us see if your answers in the previous activity is correct by showing the
graph of each linear equation in a Cartesian Coordinate plane.
The system of linear equations is represented as !
To graph, we construct table of values as learned in the previous module.
For 𝑥 + 𝑦 = 5
Assign any values to x then find y by substitution.
x
y
-2
7
If 𝑥 = −2
𝑥+𝑦 =5
(−2) + 𝑦 = 5
𝑦 =2+5
𝑦=7
0
5
2
3
If 𝑥 = 0
𝑥+𝑦 =5
(0) + 𝑦 = 5
𝑦=5
If 𝑥 = 2
𝑥+𝑦 =5
(2) + 𝑦 = 5
𝑦 = −2 + 5
𝑦=3
Thus, ordered pairs are (−2,7), (0,5), 𝑎𝑛𝑑(2,3)
For 3𝑥 + 𝑦 = 9
Assign any values to x then find y by substitution.
x
y
If 𝑥 = −2
3𝑥 + 𝑦 = 9
3(−2) + 𝑦 = 9
−6 + 𝑦 = 9
𝑦 =6+9
𝑦 = 15
-2
15
0
9
2
3
If 𝑥 = 0
3𝑥 + 𝑦 = 9
3(0) + 𝑦 = 9
0+𝑦 =9
𝑦=9
If 𝑥 = 2
3𝑥 + 𝑦 = 9
3(2) + 𝑦 = 9
6+𝑦 =9
𝑦 = −6 + 9
𝑦=3
Thus, ordered pairs are (−2,15), (0,9), 𝑎𝑛𝑑(2,3)
We are now ready to graph these lines in the coordinate plane.
The point of intersection is at (2,3).
Thus, the solution to the system
x+y=5
!
3x + y = 9
is (2,3) also which means that
𝑥 = 2 𝑎𝑛𝑑 𝑦 = 3.
To check if the ordered pair is really
the solution to the system, substitute
the 𝑥 and 𝑦 values of the (2,3) into
the equations 𝑥 + 𝑦 = 5 and 3𝑥 + 𝑦 = 9.
𝑥+𝑦 =5
(2) + (3) = 5
5=5
3𝑥 + 𝑦 = 9
3(2) + 3 = 9
6+3=9
9=9
Activity 2.
𝐓𝐑𝐔𝐄
𝐓𝐑𝐔𝐄
SOLUTION OR NOT?
Directions: Check whether the given ordered pair is a solution to the system. Write S if yes and NS if
not. Number 1 is given as an example.
1. (-1,1); !
x+y=0
x − y = -2
S
Solution: 𝑥 + 𝑦 = 9
(−1) + (1) = 0
0 = 0 𝑇𝑅𝑈𝐸
2. (2,3); !
x + 3y = 11
-2y + 5x = 4
x − 3y = -21
3. (-3,6); !-4x + 2y = 22
4. (5,-2); !
5. (0,4); !
-2x − 2y = -6
3x + 4y = 8
5x + 2y = 8
x − 5y = -20
𝑥 − 𝑦 = −2
(−1) − (1) = −2
−2 = −2 𝑇𝑅𝑈𝐸
There are three types of systems of linear equations in two variables, and three types of solutions.
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An independent system has exactly one solution pair (x, y). The point where the two lines
intersect is the only solution.
An inconsistent system has no solution. Notice that the two lines are parallel and will never
intersect.
A dependent system has infinitely many solutions. The lines are coincident. They are the
same line, so every coordinate pair on the line is a solution to both equations.
Below is a comparison of graphical representations of each type of system.
Graphing Using the Intercepts
Example 1
x + y = -1
Let us graph the system !
by using their intercepts. Take note that you need at least two points
x + y = -3
to draw a line.
From the given, both equations are written in the standard form.
To find the 𝑥 – intercept, you let 𝑦 = 0; and to find the 𝑦 – intercept, you let 𝑥 = 0.
Equation #1: 𝒙 + 𝒚 = −𝟏
𝒙 – intercept
Let 𝑦 = 0
𝑥 + 𝑦 = −1
𝑥 + 0 = −1
𝑥 = −1
𝒚 – intercept
Let 𝑥 = 0
𝑥 + 𝑦 = −1
0 + 𝑦 = −1
𝑦 = −1
The corresponding point is (-1,0).
The corresponding point is (0,-1)
Equation #2: 𝒙 + 𝒚 = −𝟑
𝒙 – intercept
Let 𝑦 = 0
𝑥 + 𝑦 = −3
𝑥 + 0 = −3
𝑥 = −3
𝒚 – intercept
Let 𝑥 = 0
𝑥 + 𝑦 = −3
0 + 𝑦 = −3
𝑦 = −3
The corresponding point is (-3,0).
The corresponding point is (0,-3)
For equation #1, plot the points (-1,0) and (0,-1)
on the coordinate plane and draw the line. While for
equation #2, use the points (-3,0) and (0,-3).
x + y = -1
Notice that the graph of the system !
x + y = -3
are parallel lines which has no solution.
This system is called Inconsistent System.
x+y=2
Example 2: Find the solution for the system !
2x = -2y + 4
Equation #1: 𝒙 + 𝒚 = 2
𝒙 – intercept
Let 𝑦 = 0
𝑥+𝑦 =2
𝑥+0=2
𝑥=2
𝒚 – intercept
Let 𝑥 = 0
𝑥+𝑦 =2
0+𝑦 =2
𝑦=2
The corresponding point is (2,0).
The corresponding point is (0,2)
Equation #2: 2𝑥 = −2𝑦 + 4
𝒙 – intercept
Let 𝑦 = 0
2𝑥 = −2𝑦 + 4
2𝑥 = −2(0) + 4
2𝑥 = 4
𝑥=2
𝒚 – intercept
Let 𝑥 = 0
2𝑥 = −2𝑦 + 4
2(0) = −2𝑦 + 4
0 = −2𝑦 + 4
2𝑦 = 4
𝑦=2
The corresponding point is (2,0).
The corresponding point is (0,2)
For equation #1, plot the points (2,0) and (0,2)
on the coordinate plane and draw the line. Same for
equation #2, use the points (2,0) and (0,2).
x + y = -1
Notice that the graph of the system !
x + y = -3
are coinciding lines which means there are
infinite number of solutions.
This is called Consistent and Dependent System.
Solving Systems of Linear Equations in Two Variables
Aside from graphing, there are alternative ways by which systems of linear equations can be solved.
Activity 3.
CAN WE MEET?
Directions: Find the solution of the system of linear equations by graphing, substitution, and elimination.
Determine if the system is consistent and independent, consistent and dependent, or inconsistent.
Write your answer in the boxes below. You may also use GeoGebra or Desmos to verify your answer.
1. !
x + y = -7
y=x+1
By Substitution Method
By Graphing
By Elimination Method
By Substitution Method
2. !
3x + y = 2
2y = 4 − 6x
By Graphing
By Elimination Method
By Substitution Method
3. !
2x − 3y = 5
3y = 10 + 2x
By Graphing
By Elimination Method
DEEPEN
In this section, you will look at some real life situations where concepts of systems
of linear equations in two variables can be applied.
Activity 4.
SOLVE THEN DECIDE!
Instruction: Anwer each of the following questions. Show your complete solutions and
explanations/justifications.
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TRANSFER
Your goal in this section is to apply your learning to real-life situations. You will be given a
practical task which will demonstrate your understanding.
Activity 5.
MY OWN EMOJI LOGIC PUZZLE
Direction: Make your own emoji logic puzzle. You can use Canva or similar apps to create your own design.
Find the solution to the puzzle through Systems of Linear Equations in Two Variables using graphing method,
substitution method, and elimination method. Then tell whether the system is consistent and independent,
consistent and dependent, and inconsistent.
Criteria for Scoring:
• Originality (25%) – The product shows a large amount of original thought. Ideas are creative.
• Accuracy of the Graph and Solutions (50%) – The puzzle is correctly translated into systems of
linear equations. The solutions and graph are exact.
• Neatness and Attractiveness (25%) – Exceptionally well designed, neat, and attractive. Colors
and design that go well together are used to make the puzzle interesting.
ANSWER KEY
Activity 1: EMOJI LOGIC PUZZLE
“Sunglasses Smile” = 2 and “Tongue Out Smile” = 3
Activity 3: TREASURE HUNT
2.
3.
Activity 4: SOLVE THEN DECIDE!
1. For short distance of travel, LG’s Rent a Car is more economical.
For long distance travel, Rent and Drive is more economical.
2. a. Php26,000 – cost of PC tablet with 12% commission
Php16,000 – cost of PC tablet with 8% commission
b. Php3,120 for PC tablet with 12% commission
Php1,280 for PC tablet with 8% commission
3. a. 120 chicken sandwiches
300 egg sandwiches
REFERENCES
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DepEd Module Mathematics 8, Systems of Linear Equations
DepEd MELC
PEAC Module Mathamatics 8
Practical Math 8 by Urgena, JNA and Canlapan, RB
https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-systems-oflinear-equations-two-variables
https://richardoco.weebly.com/uploads/1/9/7/2/19725327/module_5.pdf