What is an algebraic expression?
What are polynomials?
What is an equation?
Ans:- Two algebraic expressions or two polynomials that are combined with equal sign is called equation.
Ex:- 5xy + 9 = 3x + 4y
What is an inequation?
Ans:- Two algebraic expressions or two polynomials that are combined with greater or smaller sign is called inequation.
Ex:- 5xy + 9 > 3x + 4y,
5y - 6<2xy
What is a linear equation?
Ans:- An equation which have one and only one type of variable in each term and each term has degree one is called linear equation.
What is linear equation in one variable?
Ans:- An equation that is in the form of ax + b = 0, where a ≠ 0 is called linear equation in one variable.
The standard form of linear polynomial is ax + b = 0
What is linear equation in two variable?
Ans:- An equation that is in the form of ax + by + c = 0 where a ≠ 0 and b ≠ 0 is called linear equation in two variable.
The standard form of linear polynomial in two variable is ax + by + c = 0
At a time one and only one coefficient of linear polynomial in two variable becomes zero.
If a = 0 than b ≠ 0
If b = 0 than a ≠ 0
What is solution?
Ans:- Variable का वह मान जिसे equation मे variable के स्थान पर रखने पर equation के LHS का मान equation के RHS के मान के बराबर हो जाये , तो variable के उस मान को solution कहते हैं।
👉 linear equation का एक solution होता है।
What is solution of a linear equation in two variables?
Ans:- x = ∝ and y = β is a solution of ax +by + c = 0, if a∝ +bβ + c = 0
or
The value of x and y that satisfies the equation is called the solution of a linear equation in two variables.
Ex:- x + y = 8
x = 1, y = 7 is a solution of given equation
x = 2, y = 6 is a solution of given equation
x = 6, y = 2 is a solution of given equation
x = 7, y = 1 is a solution of given equation
x = 5, y = 3 is a solution of given equation
Note:- There are infinite solutions of linear equation in two variable
What are simultaneous of linear equations in two variables?
Or
What are pair of linear equations in two variables?
Or
What are system of linear equations in two variables?
Ans:-
Two linear equations in two unknowns are said to form a system of
simultaneous linear equations if we analyse them together and the
variables used in both the equations are same.
Ex:- x + y = 10
x - y = 1
Standard unit of system of linear equations in two variables is
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
What is solution of a system of two simultaneous linear equations?
Ans:- A pair of values of the variables x and y satisfying both of the equations in a given system of 2 simultaneous linear equations than x and y is called a solution of the system of linear equations.
Note:- There are unique solution or infinite solutions or no solutionof pair of linear equations in two variable
How many types of pair of linear equations in two variable on the basis of solution?
Ans:- There are two types of pair of linear equations in two variable on the basis of solution.
1- consistent system
2- in-consistent system
What is consistent system pair of linear equations in two variable?
Ans:- A pair of linear equations which consists at least one solution is called consistent system pair of linear equations in two variable.
What is in-consistent system pair of linear equations in two variable?
Ans:- A pair of linear equations which consists no solution is called in-consistent system pair of linear equations in two variable.
Graphical representation of pair of liner equation in two variable ( Nature of solution )
The two lines that are drawn through pair of linear equations in two variable and both lines have a common point. These type of equations have unique solution. It is also called consistent system
The two lines that are drawn through pair of linear equations in two
variable and both lines have infinite common points. These type of equations
have infinite solutions. It is also called consistent system
The two lines that are drawn through pair of linear equations in two
variable and both lines are parallel to each other. These type of equations
have no solution. It is also called in-consistent system
Standard unit of system of linear equations in two variables is
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
if ratio of a1 / a2 ≠ ratio of b1 / b2 than they have unique solution
Its converse is also true.
if a1 / a2 = b1 / b2 = c1 / c2 than they have infinite solution.
Its converse is also true.
if a1 / a2 = b1 / b2 ≠ c1 / c2 than they have no solution.
Its converse is also true.
कैसे हम बतलाएंगे की linear equation in two variables से खींची जाने वाली लाइन x-axis और y-axis को कहाँ पर काटेगा
Ans:-
step 1- equation को ax + by = c के form में लिख लें
step 2- c हमेसा +ve होना चाहिए
step 3- अब x के coefficient का symbol और y के coefficient का symbolबतलाएगा की linear equation in two variables से खींची जाने वाली लाइन x-axis और y-axis को कहाँ पर काटेगा
कैसे हम बतलाएंगे की linear equation in two variables से खींची जाने वाली लाइन origin से होकर गुजरेगा ?
Ans:- यदि equation में c का value जीरो हो , तब linear equation in two variables से खींची जाने वाली लाइन origin से होकर गुजरेगा
ALGEBRAIC METHODS OF SOLVING SIMULTANEOUS LINEAR EQUATIONS IN TWO VARIABLES:
There are three common methods that are used to solve simultaneous linear equations in two variable by algebraic method.
1- Method of elimination by substitution (or) substitution method
2- Method of elimination by equating the coefficients ( or ) elimination method
3- Method of cross multiplication
Method of elimination by substitution (or) substitution method
In
this method , we express one of the variables in terms of the other
variable from either of the two equations and then this expression is
put in the other equation to obtain an equation in one variable .
For
solving simultaneous linear equations in two variable by Method of
elimination by substitution (or) substitution method, We should follow
the following steps.
Step1:- Express Y in terms of X in one of the given equation.
Step2:- Substitute this value of Y in term of X in the other equation. This gives a linear equation in X\
Steo3:- solve the linear equation in X. obtain the value of X
Step4:- Substitute this value of X in the relation taken in step 1 to obtain a linear equation in Y.
Step5:- Solve the above linear equation in Y to get the value of Y
Method of elimination by equating the coefficients ( or ) elimination method
In this method eliminate one of the 2 variables to obtain an equation in one variable which can be solved easily. Now put the value of this variable in any one of the given equation, The value of the other variable can be obtained
For
solving simultaneous linear equations in two variable by Method of
elimination by equating the coefficients elimination method, We should follow
the following steps.
Step1:- multiply the given equation by suitable numbers so as to make the coefficients of one of the unknown variables numerically equal.
step2- If the numerically equal coefficients are opposite in sign then add the new equation. otherwise subtract them.
step3- resulting equation is linear in one unknown variable.
step4- Substitute this value in any of the given equations.
Step5- Solve it to get the value of the other unknown variable.
Step3-
