Systems of Linear Equations
in Two Variables
Activity: 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?
Systems of Linear Equations
in Two Variables
 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
 The system of linear equations is represented as x + y = 5 and 3x + y = 9 . The brace
denotes that the two equations can
 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.
To graph our previous activity each linear equation in a Cartesian Coordinate plane.
Lets construct table:
For 𝑥 + 𝑦 = 5
Assign any values to x then find y by substitution.
x
y
-2
0
2
For 3𝑥 + 𝑦 = 9
Assign any values to x then find y by substitution.
x
y
-2
0
2
What is the ordered pairs of x+y=5 and
3x+y=9
For 𝑥 + 𝑦 = 5
The ordered pairs are (−2,7), (0,5), 𝑎
𝑛𝑑(2,3)
For 3𝑥 + 𝑦 = 9
Thus, ordered pairs are (−2,15),
(0,9), 𝑎𝑛𝑑(2,3)
We are now ready to graph these lines in the coordinate plane.
Activity 2: 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
Solution: 𝑥 + 𝑦 = 9
(−1) + (1) = 0
S
𝑥 − 𝑦 = −2
(−1) − (1) = −2
0 = 0 𝑇𝑅𝑈𝐸−2 = −2 𝑇𝑅𝑈𝐸
2. (2,3); ! x + 3y = 11
-2x + 5X = 4
3. (-3,6); ! x − 3y = -21
-4x + 2y = 22
4. (5,-2); !-2x − 2y = -6
3x + 4y = 8
5. (0,4); ! 5x + 2y = 8
x − 5y = -20
There are three types of systems of linear
equations in two variables, and three types
of solutions.
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.
Graphing Using the Intercepts
 Example 1
 Let us graph the system x + y = -1 by using their intercepts. Take note that
you need at least two points 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
𝒚 – intercept
Let 𝑥 = 0
What is the corresponding point of x and y – intercept?
 Equation #2: 𝒙 + 𝒚 = −𝟑
𝒙 – intercept
 Let 𝑦 = 0
𝒚 – intercept
Let 𝑥 = 0
What is the corresponding point of x and y – intercept?
 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).
Notice that the graph of the system ! x + y = -1
x + y = -3
are parallel lines which has no solution.
This system is called Inconsistent System.
 Example 2: Find the solution for the system ! x + y = 2
2x = -2y + 4
Equation #1: 𝒙 + 𝒚 = 2
𝒙 – intercept
Let 𝑦 = 0
𝒚 – intercept
Let 𝑥 = 0
Equation #2: 2𝑥 = −2𝑦 + 4
𝒙 – intercept
𝒚 – intercept
Let 𝑦 = 0
Let 𝑥 = 0
What is the corresponding point of x and y – intercept?
 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).
 Notice that the graph of the system x + y
= 2 and 2x = -2y + 4 are coinciding lines
which means there are infinite number of
solution.
What is the types of system of linear equations in
two variables?
This is called Consistent and Dependent System.
Solving Systems of Linear Equations in
Two Variables
This is the alternative ways by which systems of linear
equations can be solved.
There are two alternative ways solving a linear
equation.
1. Substitution Methods
2. Elimination Methods
To solve a system of linear equation by
substitution method, the following
procedures could be followed:
a. Solve for one variable in terms of the other variable in one
of the equations. If one of the equations already gives the
value of one variable, you may proceed to the next step.
b. Substitute to the second equation the value of the
variable found in the first step. Simplify then solve the
resulting equation.
c. Substitute the value obtained in (b) to any of the original
equations to find the value of the other variable.
d. Check the values of the variables obtained against the
linear equations in the system.
Example 1:
 Solve the system 2x + y = 5 and –x + 2y = 5 by substitution method.
To solve a system of linear equation by
elimination method, the following
procedures could be followed:
A. Whenever necessary, rewrite both equations in standard form Ax + By = C
B. Whenever necessary, multiply either equation or both equations by a nonzero
number so that the coefficients of x or y will have a sum of 0. (Note: the
coefficients of x and y are addictive inverses.)
C. add the resulting equations. This leads to an equation in one variable. Simplify
then solve the resulting equation.
D. Substitute the value obtained to any of the original equations to find the
value of the other variable.
E. Check the values of the variables obtained against the linear equations in the
system.
Example: Solve the system 3x + y = 7
and 2x -5y =16 by elimination method.
Activity 3:
 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.
1. x + y = -7 and y = x + 1
2. 3x + y = 2 and 2y = 4 − 6x
3. 3x - 2y = 5 and 3y = 10 + 2x
Activity 4: SOLVE THEN DECIDE!
 Instruction: Answer each of the following questions. Show your
complete solutions and explanations/justifications.
Assignment: 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.