The document discusses linear equations in two variables. It defines linear equations as equations containing two variables where each variable has an exponent of 1. It provides examples and discusses the general form of simultaneous linear equations as a1x + b1y = c1 and a2x + b2y = c2. The document also discusses framing linear equations from word problems, graphically representing solutions, criteria for consistent/inconsistent systems, and methods for algebraically solving simultaneous linear equations including elimination, substitution, and cross multiplication.
AN EQUATION WHICH CAN BE WRITTEN IN THE FORM OF ax+by+c=0 WHERE a,b and c ARE REAL NUMBERS.
YOU WILL GET TO KNOW HOW TO REPRESENT THE EQUATIONS IN A GRAPH.
Identify basic properties of equations
Solve linear equations
Identify identities, conditional equations, and contradictions
Solve for a specific variable (literal equations)
AN EQUATION WHICH CAN BE WRITTEN IN THE FORM OF ax+by+c=0 WHERE a,b and c ARE REAL NUMBERS.
YOU WILL GET TO KNOW HOW TO REPRESENT THE EQUATIONS IN A GRAPH.
Identify basic properties of equations
Solve linear equations
Identify identities, conditional equations, and contradictions
Solve for a specific variable (literal equations)
Algebra is used in many field in many different ways to solve equation problems, and in business algebra is also used or in our day to day life it is also used. ... Algebra is a way of keeping track of unknown values, which can be used in equations.
21 - GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES.pptxbernadethvillanueva1
GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES
GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES
GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES
GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
This will help you on how to solve quadratic equations by factoring.
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Algebra is used in many field in many different ways to solve equation problems, and in business algebra is also used or in our day to day life it is also used. ... Algebra is a way of keeping track of unknown values, which can be used in equations.
21 - GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES.pptxbernadethvillanueva1
GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES
GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES
GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES
GRAPHS THE SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
This will help you on how to solve quadratic equations by factoring.
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
Order of presentation
Anushka - Opening
Nikunj -Intro
Shubham - Graphical
Amel - Sunstitution
Siddhartha- Elimination
Karthik - Cross multiplication
Anushka - Equations reducible...& wrap-up
In case of any confusion..inform me by facebook, phone or in school
Power Point Presentation on a PAIR OF LINEAR EQUATION IN TWO VARIABLES, MATHS project...
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1. LINEAR EQUATIONS
IN TWO VARIABLES
BY: DR. VIVEK NAITHANI
TGT MATHS
KENDRIYA VIDYALAYA SANGATHAN
Copyright Information: CC by SA 4.0
2. DEFINITION:
An equation that contains two variables and the degree of
each variable being 1 is called linear equation in two
variables.
e.g. : 2x+3y =7
3x- 6y =34
A system of linear equations always exists in pair of
equations.
3. GENERAL LINEAR EQUATION IN TWO
VARIABLES
The general form of simultaneous linear equations is:
a1x + b1y = c1 ------------------- (1)
a2x + b2y = c2 -------------------- (2)
Here a1, a2 are the co-efficients of x
b1, b2 are the co-efficients of y
c1, c2 are the constants.
Note: Both a & b together are never 0
4. FRAMING A LINEAR EQUATION IN TWO
VARIABLES
To frame a linear equation in two variables working
rule is:
1. Select the unknown and assign variables to the unknown.
2. Select the given condition and frame the variables in the given
first and second conditions.
3. Frame the equations.
5. EXAMPLE:
Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also,
three years from now, I shall be three times as old as you will be.” Represent this situation
algebraically.
Here two unknowns are age of Aftab and age of daughter.
Let present age of Aftab be x and present age of daughter be y.
I condition: 7 years ago Aftab was 7 times as old as his daughter
7 years ago age if Aftab = (x-7) and age of his daughter = (y-7) years
∴ (x-7) = 7 X (y-7)
⇒ x- 7 = 7y -49 ⇒ x – 7y = -42
x – 7y = -42 This is first equation in two variables.
6. II Condition: Three years from now, Aftab shall be three times as
old as his daughter will be.
After 3 years: Aftab’s age = x+3 Daughter’s age = y+3
⇒ (x+3) = 3 X (y+3)
⇒ x+3 = 3y +9
⇒ x -3y = 6 This is second linear equation
Hence the set of linear equation in two variables is
x – 7y = -42 ---------------(1)
x -3y = 6 ---------------------(2)
7. GRAPHICAL REPRESENTATION OF SIMULATENOUS LINEAR
EQUATIONS
• A pair of simultaneous linear equation represents a pair of straight lines on a graph sheet.
8. CRITERIA OF CONSISTENCY OF SOLUTION OF
SIMULTANEOUS LINEAR EQUATIONS
For a given set of pair of linear equations in two variables:
a1x + b1y = c1 ------------------- (1)
a2x + b2y = c2 -------------------- (2)
The following cases arise as solution of the given set of
equations:
1. CASE I: If
𝑎1
𝑎2
≠
𝑏1
𝑏2
then the system is consistent and
has a
unique solution.
12. ALGEBRAIC SOLUTION OF SIMULATNEOUS
LINEAR EQUATIONS
The following methods are being used to solve simultaneous linear equations
algebraically:
1.Elimination Method.
2.Substitution Method.
3.Cross multiplication method.
13. ELIMINATION METHOD:
In this method we eliminate one of the two variables to achieve the
desired results
Solve the equations 2x+y = 5 and 3x+2y =8.
2x+y = 5 --------(1) X 3 [ Here we multiply the equations with the suitable non-zero
3x+2y = 8 -------(II) X 2 constants to make co-efficients of the variable to be
eliminated equal.]
So our equations become 6x +3y = 15 {Here we are eliminating x and now
6x +4y = 16 subtracting the new equations we get}
-y = -1 ⇒ y = 1
Substituting y =1 in any of the equations (Here we are substituting in eqn. 1) we get 2x
+1= 5
⇒ 2x = 4 ⇒ x= 2.
14. SUBSTITUTION METHOD:
STEP 1: In this method we first choose an equation and then express
any one of the two variables in terms of the other.
STEP 2: Then we substitute the function obtained in step 1 in the another
equation.
STEP 3: Now we get an equation in a single variable which on solving will
give value of one variable.
STEP 4: The value obtained in step 3 will be substituted in the another
equation to get the value of another variable.
15. EXAMPLE
Solve the equations 2x+y = 5 and 3x+2y =8.
2x+y = 5 ⇒ y = 5- 2x { Here variable y has been expressed in terms of
x}
Substituting y = 5-2x in the second equation we get
3x + 2(5-2x) = 8 ⇒ 3x + 10 -4x = 8
⇒ 3x-4x = 8-10 ⇒ -x = -2 ⇒ x = 2
Now substituting x= 2 in y = 5-2x we get
y = 5 – 2(2) = 5-4 = 1
Hence the solution is x = 2 and y = 1.
16. CROSS MULTIPLICATION METHOD:
The given equations are a1x+b1y + c1= 0 and a2x+b2y+c2= 0
a1x+b1y + c1= 0
a2x+b2y+c2 = 0
For solving this system of equation we follow a pattern given in following diagram
17. Hence we get the following result:
Solving this first and third fractions we will get the
value of x and solving second and third fractions we
will get the value of y.
18. Solve the following equations
x-3y-7=0 and 3x-3y-15=0
x-3y-7 = 0
3x-3y-15= 0
𝒙
−𝟑 𝑿 −𝟏𝟓 − −𝟑 𝑿(−𝟕)
=
𝒚
𝟑 𝑿 −𝟕 − 𝟏 𝑿(−𝟏𝟓)
=
𝟏
𝟏 𝑿 −𝟏𝟓 − 𝟑 𝑿(−𝟕)
Which gives
𝑥
45 − 21
=
𝑦
−21 + 15
=
1
−15 + 21
𝑥
24
=
𝑦
−6
=
1
6
Solving fraction 1 and 3 we get x = 24/6 = 4
Solving fraction 2 and 3 we get y = -6/6 = -1. Hence Solution is x= 4 and y = -1