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Page 1 A pair of linear equation in two Variables Page 2 A pair of linear equation in two Variables Substitution Method Linear equation can be represented and solved by Equation in the form of ax + by + c = 0, where a, b & c are real numbers and ‘a’ & ‘b’ both cannot be zero is known as linear equation in two variables Graphical Method Algebraic Method Substitution Elimination Cross Multiplication Cross multiplication method Elimination Method . We nd the value of one variable ( say y ) in terms of the other variable ( say x ) from either of the equations. . Substitute the value of y in another equation to make it in one variable. Now we can get the value of x. . Substitute the value of obtained variable x in any of the equations to get the value of ‘y ’. . We make coecient of a variable in one of the equations equal to the coecient of the same variable in another equation by multiplying it with a suitable number. . Add or subtract one equation from other to eliminate one variable. . Now we get an equation in one variable and can solve it to get its value. . Substitute the value of the obtained variable in any of the equations to get the value of another variable. 1 1 1 2 3 2 3 4 . Let the general from of pair of linear equations in two variables x & y be a 1 x + b 1 y + c 1 = 0 a 2 x + b 2 y + c 2 = 0 2 . x & y can be found by using following formulas b 1 c 2  b 2 c 1 a 1 b 2  a 2 b 1 c 1 a 2  c 2 a 1 a 1 b 2  a 2 b 1 y = x = Methods to solve pair of linear equations A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 7 A Pair of Linear Equations in Two Variables { Sotution } Y X 1 1 (0,0) 2 x + y = 6 x + y = 4 (2, 2) x=2, y=2 2 2 3 3 4 4 5 5 6 6 7 7 8 9 . Page 3 A pair of linear equation in two Variables Substitution Method Linear equation can be represented and solved by Equation in the form of ax + by + c = 0, where a, b & c are real numbers and ‘a’ & ‘b’ both cannot be zero is known as linear equation in two variables Graphical Method Algebraic Method Substitution Elimination Cross Multiplication Cross multiplication method Elimination Method . We nd the value of one variable ( say y ) in terms of the other variable ( say x ) from either of the equations. . Substitute the value of y in another equation to make it in one variable. Now we can get the value of x. . Substitute the value of obtained variable x in any of the equations to get the value of ‘y ’. . We make coecient of a variable in one of the equations equal to the coecient of the same variable in another equation by multiplying it with a suitable number. . Add or subtract one equation from other to eliminate one variable. . Now we get an equation in one variable and can solve it to get its value. . Substitute the value of the obtained variable in any of the equations to get the value of another variable. 1 1 1 2 3 2 3 4 . Let the general from of pair of linear equations in two variables x & y be a 1 x + b 1 y + c 1 = 0 a 2 x + b 2 y + c 2 = 0 2 . x & y can be found by using following formulas b 1 c 2  b 2 c 1 a 1 b 2  a 2 b 1 c 1 a 2  c 2 a 1 a 1 b 2  a 2 b 1 y = x = Methods to solve pair of linear equations A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 7 A Pair of Linear Equations in Two Variables { Sotution } Y X 1 1 (0,0) 2 x + y = 6 x + y = 4 (2, 2) x=2, y=2 2 2 3 3 4 4 5 5 6 6 7 7 8 9 . A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 8 If Condition of Consistency Number of Solutions of a Pair of Linear Equations Pair of Linear Equations { Lines are intersecting } { Unique solution } { Consistent } Y X 1 1 (0,0) 2 m + a = 6 m + a = 4 (2, 2) 2 2 3 3 4 4 5 5 6 6 7 7 8 9 . {Lines are parallel} {No solution} {Inconsistent} Y X 1 1 (0,0) m + a = 6 m + a = 4 2 2 3 3 4 4 5 5 6 6 7 7 8 9 = = a 1 a 2 b 1 b 2 c 1 c 2 {Lines are coincident} {Many solutions} {Consistent} Y X 1 1 (0,0) 2 2 3 3 4 4 5 5 6 6 7 7 8 9 2m + 2a = 8 m + a = 4 = = a 1 a 2 b 1 b 2 c 1 c 2 a 1 a 2 = b 1 b 2 a 1 a 2 = b 1 b 2 Number of Solutions Relation Between Lines (Inconsistent) No Solution (Inconsistent) No Solution Coincident Lines Parallel Lines Parallel Lines Intersecting Lines Intersecting Lines (Consistent) Unique Solution (Consistent) Many Solution (Consistent) Unique Solution { Then check for } 2x + 4y = 6 4x  8y = 12 ; x  3y = 1 2x  6y = 4 ; 2p  3q = 2 3p  2q = 2 ; m + 4n = 1 3 2 3m + 8n = 2 ; 6x + 2y = 3 3x + y = 3 ; Page 4 A pair of linear equation in two Variables Substitution Method Linear equation can be represented and solved by Equation in the form of ax + by + c = 0, where a, b & c are real numbers and ‘a’ & ‘b’ both cannot be zero is known as linear equation in two variables Graphical Method Algebraic Method Substitution Elimination Cross Multiplication Cross multiplication method Elimination Method . We nd the value of one variable ( say y ) in terms of the other variable ( say x ) from either of the equations. . Substitute the value of y in another equation to make it in one variable. Now we can get the value of x. . Substitute the value of obtained variable x in any of the equations to get the value of ‘y ’. . We make coecient of a variable in one of the equations equal to the coecient of the same variable in another equation by multiplying it with a suitable number. . Add or subtract one equation from other to eliminate one variable. . Now we get an equation in one variable and can solve it to get its value. . Substitute the value of the obtained variable in any of the equations to get the value of another variable. 1 1 1 2 3 2 3 4 . Let the general from of pair of linear equations in two variables x & y be a 1 x + b 1 y + c 1 = 0 a 2 x + b 2 y + c 2 = 0 2 . x & y can be found by using following formulas b 1 c 2  b 2 c 1 a 1 b 2  a 2 b 1 c 1 a 2  c 2 a 1 a 1 b 2  a 2 b 1 y = x = Methods to solve pair of linear equations A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 7 A Pair of Linear Equations in Two Variables { Sotution } Y X 1 1 (0,0) 2 x + y = 6 x + y = 4 (2, 2) x=2, y=2 2 2 3 3 4 4 5 5 6 6 7 7 8 9 . A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 8 If Condition of Consistency Number of Solutions of a Pair of Linear Equations Pair of Linear Equations { Lines are intersecting } { Unique solution } { Consistent } Y X 1 1 (0,0) 2 m + a = 6 m + a = 4 (2, 2) 2 2 3 3 4 4 5 5 6 6 7 7 8 9 . {Lines are parallel} {No solution} {Inconsistent} Y X 1 1 (0,0) m + a = 6 m + a = 4 2 2 3 3 4 4 5 5 6 6 7 7 8 9 = = a 1 a 2 b 1 b 2 c 1 c 2 {Lines are coincident} {Many solutions} {Consistent} Y X 1 1 (0,0) 2 2 3 3 4 4 5 5 6 6 7 7 8 9 2m + 2a = 8 m + a = 4 = = a 1 a 2 b 1 b 2 c 1 c 2 a 1 a 2 = b 1 b 2 a 1 a 2 = b 1 b 2 Number of Solutions Relation Between Lines (Inconsistent) No Solution (Inconsistent) No Solution Coincident Lines Parallel Lines Parallel Lines Intersecting Lines Intersecting Lines (Consistent) Unique Solution (Consistent) Many Solution (Consistent) Unique Solution { Then check for } 2x + 4y = 6 4x  8y = 12 ; x  3y = 1 2x  6y = 4 ; 2p  3q = 2 3p  2q = 2 ; m + 4n = 1 3 2 3m + 8n = 2 ; 6x + 2y = 3 3x + y = 3 ; Variable Reducible method + = 4 (x  y) 1 = p = q (x  y) 1 (x + y) 3 (x + y) 2 Take , ? 4p + 3q = 2 + = 6 y 1 = p = q x 1 y 3 x 4 Take Divide by x? , ? ? 6x + 3y = 4 x? 6p + 3q = 4 ? ? + = 2 m 7 Take n 3 2x + 7y = 3 ? = , 1 m 1 n x = y Introduction to Pair of Linear Equations Problem Solving of Interesting Lines Conditions of Consistency Scan the QR Codes to watch our free videos PLEASE KEEP IN MIND ? ? ? ? Linear means straight and a linear equation refers to an equa tion which when plotted on the graph gives a straight line. And it is true for any linear equation whether it is one variable, two variable or multivariable. To solve linear equations, number of variables should be equal to the number of equations. If it is asked to solve linear equations and the method is not mentioned then, one can use any of the methods which is most suita ble for that particular question. There is a formula that shows the relation between time, speed and distance. Use this formula for word problems. Speed = Distance Time There can be a variety of situations which can be mathematically represented by two equations which are not linear. But we can eventually alter them and reduce them to a pair of linear equations. A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 9Read More
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1. What are the types of solutions for a pair of linear equations in two variables? 
2. How can we solve a pair of linear equations using the substitution method? 
3. What is the graphical representation of a pair of linear equations with no solution? 
4. How can we determine the number of solutions for a pair of linear equations using the determinant? 
5. Can a pair of linear equations have more than one unique solution? 

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