The document discusses linear equation systems with 2 and 3 variables. It provides examples of linear equations with 1, 2, and 3 variables. There are 4 methods to solve a system of 2 linear equations: substitution, elimination, elimination-substitution, and graphing. Graphing involves drawing the lines represented by each equation and finding their point of intersection. For 3 variables, one variable is eliminated to obtain 2 equations, which are then solved to find the values for the remaining 2 variables. These values are then substituted back into one of the original equations to solve for the eliminated variable.

1. LINEAR EQUATION SYSTEM
with 2 and 3 variables
FIRST MEETING

2. DO YOU REMEMBER ?
• Do you still remember about linear equation of one
variable? Can you give some examples, students?
• Which one of the following example is linear
equation of one variable? Give your reason.
2
1
a) (4 2) 3
2
b) 3 2 5 0
c) 3 2 7 2
d) 2 3
x
x x
y y
x y
+ =
+ − =
− = −
− =
The point (a) and (c) are
examples of linear
equation of one variable.
Can you find the
solution?

3. Instructional Goals
Students are able to determine solution of
linear equation system with 2 variables using
graph, substitution, elimination and
elimination-substitution.
Students are able to determine solution of
linear equation system with 3 variables

4. • Which one is linear equation with 2
variables?
• Can you give another example?
2
1
a) (4 2) 3
2
b) 3 2 5 0
c) 3 2 7 2
d) 2 3
x
x x
y y
x y
+ =
+ − =
− = −
− =

5. • If I have
what the meaning of solution of that system?
• In how many ways we can solve linear equation
system?
we can solve linear equation system in four
ways, that are substitution, elimination,
substitution-elimination and graph method
2 4
2 3 12
x y
x y
− =
+ =

6. Linear Equation Sytem with 2 Variables
• In general, a linear equation system of 2
variables x and y can be expressed:
• Let’s try to solve this problem
1 1 1
2 2 2
a x b y c
a x b y c
+ =
+ =
1 1 1 2 2 2where , , , , , anda b c a b c R∈
2 4
2 3 12
x y
x y
− =
+ =
{ }( , )SS x y=

7. Linear Equation Sytem with 2 Variables
• Graphic Method
1. Draw a line 2x – y
= 4. What is the
solution of that
linear equation?
2. Also draw a line
2x + 3y = 12. What
is the solution of
that linear
equation?

8. Linear Equation Sytem with 2 Variables
 So, can you guess the solution of that
linear equation system?
• Conclusion:
The solution of that linear system is the
point of intersection of the lines

9. Linear Equation Sytem with 3 Variables
 Can you guess the form of linear equation system of 3
variables, class?
• In general, the linear equation system in variables x, y, z can
be writeen as:
 How to solve the linear equation system of 3 variables?
1 1 1 1
2 2 2 2
3 3 3 3
= .......(1)
....(2)
....(3) ; where , , , 1,2,3i i i
a x b y c z d
a x b y c z d
a x b y c z d a b c R i
+ +
+ + =
+ + = ∈ =

10. Linear Equation Sytem with 3 Variables
Example: Solve the following system:
2 5 ........(1)
2 9 ........(2)
2 3 4 ......(3)
x y z
x y z
x y z
− + =
+ − =
− + =
Suppose we want to eliminate variable x, what should we do?
 Take eq.(1) and (2), then eliminate variable x.
We will get:
 Take eq.(1) and (3), then eliminate variable x.
We will get:
Take eq.(4) and (5), then find the value of y=y1 and
substitute to eq.(4) or(5) to get z=z1 or vice versa.
Substitute y=y1 and z=z1 to eq.(1), (2) or (3) to get x=x1.
Thus solution set is
3 5 1 .........(4)y z− + =
1 .......(5)y z− =
( ){ }1 1 1, ,x y z

11. > Students Worksheet <
1. I will make a group. Each group consist of 4
member. So please gather with your group now,
class. (STAD Model)
2. Do and discuss the worksheet I.
3. Some groups will present their result
discussion in front of the class.

12. GROUPs
• Group 1
– Ari
– Budi
– Caca
– Deni
• Group 2
• Rohmi
• Fino
• Gegi
• Hana
• Group 4
– Munir
– Atin
– Oki
– Prasda
• Group 3
– Teja
– Jorinda
– Karin
– Lely

13. TODAY’S CONCLUSION
• There are 4 ways to solve linear equation
system with 2 variables :
– Substitution
– Elimination
– Elimination-substitution
– graph method

14. TODAY’S CONCLUSION
- Using Substitution Method –
1.Write one of the equation in the form
2.Substitute y (or x) obtained in the first
step into the other equations
3.Solve the equations to obtain the value
4.Substitute the value x=x1 obtained to get y1
or substitute the value y=y1 obtained to get x1
5. The solution set is
ory ax b x cy d= + = +
1 1orx x y y= =
( ){ }1 1,x y

15. TODAY’S CONCLUSION
• Using Elimination Method –
• The procedure for eliminating variable x (or
y)
1. Consider the coefficient of x (or y). If they
have same sign, subtract the equation (1) from
equation (2), if they have different sign add
them.
2. If the coefficients are different, make them same
by multiplying each of the equations with the
corresponding constants, then do the addition or
subtraction as the first step.
1 1 1
2 2 2
............. (1)
............. (2)
a x b y c
a x b y c
+ =
+ =

16. TODAY’S CONCLUSION
• Using Graphing Method –
The solution of the linear system with two
variables is the point of intersection of the
lines.
• If 2 lines are drawn in the same coordinate
there are 3 possibilities of solution :
– The lines will intersect at exactly one point if the
gradients of the lines are different. Then the
solution is unique
– The lines are parallel if the gardients of the line
are same. Thus, there are no solution
– The line are coincide to each other if one line is a
multiple of each other. Thus the solution are
infinite

17. TODAY’S CONCLUSION
• The procedures for solving linear equation in
3 variables are:
– Eliminate one of the variables from the three
equations, say x such that we obtain a linear
equation in 2 variables. Eliminate variable x from
equation (1) and (2), label the result as equation
(4). Then eliminate variable x from equatin (1) (or
(2)) and equation (3), label the result as equation
(5). Hence equation (4) and (5) are linear system
with 2 variables y and z
– Solve the linear system to yield the values of y and
z
– Substitute the result to equation (1), (2), or (3)
to obtain x
– Write the solution set