PRESENTATION SUBJECT SANMATI HIGHER SECONDARY SCHOOL MATHEMATICS SUBMITTED TO VINOD PATIL SIR

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PRESENTATION TOPIC LINEAR EQUATION IN TWO VARIABLES

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MADE BY RAGHAV GHULE Xth A

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Two linear equations in the same two variables are called linear equation in two variables the most general form of a pair of linear equations is a 1 x+b 2 y +c 1 =0 a 2 x+b 2 y+c 2 =0 Where a 1 ,a 2 ,b 1 ,b 2 ,c 1 ,c 2 are real numbers ,such that a 1 2 +b 1 2 = 0 a 2 2 +b 2 2 =0 A pair of linear equations in two variables can be represented and solved by the : Graphical method Algebraic method Graphical method : the graph of the pair of linear equation in two variables is represented by two lines : INTRODUCTION

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If the lines intersect at a point then that point gives the unique solutions of the two equations . In this the pair of equations is consistent. If the lines coincide then there are infinitely many solutions – each point on the line being a solutions. In this case the pair of linear equation is dependent ( consistent ). 3. If the lines are parallel ,then the pair of linear equation is inconsistent . Algebraic method : There are several method for solving the equations by algebraic methods Substitution Method Elimination Method Cross Multiplication Method

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Two or more different linear equations in two variables having the same variables taken together from a system of linear equations in two variables. Thus a system of linear equation is a collection or set of all those linear equations which we deal together simultaneously. System of linear equation in two variables

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When we consider two linear equations in two variables such equation are called Simultaneous Solutions . When the simultaneous solution are satisfied with same pair of values of two variables, then such pair of values of the two variables then such pair of values of the variables is called the Solution of simultaneous equations Let 2x + y = 8 and x+ 3y = 9be the given simultaneous equation. The solution of these simultaneous equations is the pair of the x and y that satisfy both the equation simultaneously Simultaneous Solution Solutions of Simultaneous Solutions

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Graphical Method

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The graphical method is the method in which the linear equations are solved on the graph Consider these are simultaneous equations x + 2y = 0 and 3x + 4y – 20 = 0 2x + 3y - 9 = 0 and 4x + 6y - 18 = 0 x + 2y - 4 =0 and 2x + 4y -12 =0 There are three conditions after solving the linear equation through graphical method we get Intersecting Lines Coincident Lines Parallel Lines Example in the above three equations The lines will Intersect The lines will coincide The lines will parallel

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The graphs of equation x + y = 3 and x – y = 1 are two distinct the point lines which intersect each other in exactly one point ( 2 , 1 ). This point of intersection has value of x = 1 and y = 1. These equation satisfy both the equations and hence ( 2 , 1 ) is the solution of the given simultaneous equations. Therefore , the co-ordinates of the points of intersection of lines represented by these equations in the solution. 1.1 Intersecting Lines

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x + y = 3 x - y = 1

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The graphs of the equation x - y = -2 and 2x - 2y = -4 appears to in one lines but these are two lines drawn one above the other here. Here every point of the line is the intersection point. Hence all the point on the line has solutions. Thus these have infinitely many solutions these lines are overlapping ( coincident ). If the lines coincide with each other then they have infinite solutions , each point on the lines being a solution. 1.2 Co-incident Lines

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x - y = -2

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Consider the equation 2x - y = -1 and 2x – y = -4. Do they have a common solution the lines. These equations have no solution. The line of these two equations does not intersect each other. Therefore the lines represented by these equations are parallel. 1.3 Parallel Lines

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2x - y = -4 2x - y = -1

Limitations of graphical method :

Limitations of graphical method If the solutions of simultaneous equations has large values of the variables it will be the inconvenient to use graphic method. If the solutions of simultaneous equations has fractional values of the variables the accuracy will not be possible. If the angle between the two lines is small, it is not possible to find the exact point of intersection.

2 Algebraic:

2 Algebraic Methods

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Elimination method is the second method of solving linear equation by algebraic method. This method is sometime more convenient than the substitution method Step 1 :- First multiply both the equation by some suitable non-zero constants to make the co efficient of one variable either x or y Step 2 :- Then add or subtract both equation so that the on variable get eliminated if you get one variable go to step 3 If in step 2 if we obtain a true statement involving no variable then the original pair of linear equation have infinitely many solution If in step 2 we obtain a false statement involving no variable then the original pair linear equation have no solution Step 3 :- solve the equation in one variable x or y so obtained to get its value Step 4 :- Substitute the value of x in the either of original equations to get the value of the other variable 2.1 Elimination Method

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Example :- x + y = 5 1) 2x – 3y = 4 2) We will multiply equation 1) by 2 After multiplication of equation 1 we got the equation 2x + 2y = 10 3) Subtraction of equation 2) and 3) 2x + 2y = 10 - 2x – 3y = 4 5y = 6 y = By putting value of y in equation 1) x + y = 5 x + = 5 x = 5 - x = x =

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This one of the algebraic method to solve simultaneous equation in this method the equations are solved by taking value of each variable separately let s see the steps to solve the Substitution method 2.2 Substitution Method

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These are the steps to solve linear equation ( simultaneous equation ) by substitution method Step 1:- find the value of one variable, say y in terms of other , i.e. x from either equation whichever is convenient. Step 2:- substitute the value of y in the other equation and reduce it to an equation in one variable , i.e. the terms of x which can be solved Step 3:- Substitute the value of x or y obtained in step 2 in the equation used in Step 1 to obtain the value of the other variable.

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Example :- Let the two equations be 3x + 4y = 10 ; x - y = 1 x – y = 1 x = y + 1 x - y = 1 1) x = y + 1 2) 3x + 4y = 10 3) 3 ( y + 1 ) + 4y = 10 4) 3y + 3 + 4y = 10 3y + 4y + 3 = 10

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7y + 3 = 10 7y = 10 – 3 7y = 7 y = y = 1 x = y + 1 x = 1 + 1 x = 2 Here is the answer x = 2 ; y = 1

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The method of obtaining solution of simultaneous equation by using determinants is known as Cramer’s rule. In this method we have to follow this equation and diagram x y 1 b 1 c 1 a 1 b 1 b 2 c 2 a 2 b 2 2.3 Cross-multiplication method [ Cramer’s rule ]

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For solving the method of cross – multiplication method we have to follow then following steps :- Step 1 :- Write the equations in the form 1) & 2) Step 2 :- take the help of the above diagram and equation written above Step 3:- Find x and y provided a 1 b 2 - a 2 b 1 ≠ 0

Gabriel Cramer:

Gabriel Cramer Gabriel Cramer was born on 31 July 1704 in Geneva and dead on 4 January 1752. He was a Swiss Mathematician. He showed excellence in mathematics from an early age. At the age of 18 he received his doctorate. In 1728 he proposed the solution to St. Petersburg Paradox. He edited the works of two elders Bernoullis and wrote on the physical cause of the spheriodial shape of the planets in 1730 and on the Newton’s treatment of cubic curves in 1956. he was professor at Geneva. He has proposed the algebraic method for solving linear equation in two variables named cross-multiplication or Cramer’s rule.

Summary :

Summary Linear equations in two variables have infinite solutions Linear equations in two variables is expressed as ax + by + c = 0 where a ≠ 0 b ≠ 0 Two or more linear equation having same variables taken form linear equation in two variables . When we consider linear equation in two variables When we consider linear equation in two variables such equation are called Simultaneous Equation When the Simultaneous equations are satisfied by the same pair of values of the two variables then such pair of values of variables is called the solutions of given Simultaneous Equation

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Methods to solve Simultaneous Equation Graphical method Algebraic method The Elimination Method Substitution Method Cross - Multiplication Method The Simultaneous Equation are said to be consistent if they have solutions The Simultaneous Equation are said to be inconsistent if they have no solutions

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