Page 1 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Algebraic Expression: A combination of constants and variables, connected by four fundamental arithmetic operations of +, -, x and ÷ is called an algebraic expression. Equation: An algebraic expression with equal to sign (=) is called an equation. Without an equal to sign, it is considered as an expression only. Linear Equation: If the greatest exponent of the variable(s) in an equation is one, then the equation is said to be a linear equation. If number of variable used in linear equation is one, then equation is said to be linear equation in two variables. Linear Equation in Two Variables: An equation which can be put in the form 0 = + + c by ax , where a, b and c are real numbers, and 0 , 0 ? ? b a is called a linear equation in two variables x and y . General Form of a Pair of Linear Equation in Two Variables: A pair of linear equations in two variables x and y can be represented as follows: a 1 x + b 1 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 System of simultaneous linear equations: Consistent system: A system of simultaneous linear equations is said to be consistent if it has at least one solution. Inconsistent system: A system of simultaneous linear equations is said to be inconsistent if it has no solution. Page 2 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Algebraic Expression: A combination of constants and variables, connected by four fundamental arithmetic operations of +, -, x and ÷ is called an algebraic expression. Equation: An algebraic expression with equal to sign (=) is called an equation. Without an equal to sign, it is considered as an expression only. Linear Equation: If the greatest exponent of the variable(s) in an equation is one, then the equation is said to be a linear equation. If number of variable used in linear equation is one, then equation is said to be linear equation in two variables. Linear Equation in Two Variables: An equation which can be put in the form 0 = + + c by ax , where a, b and c are real numbers, and 0 , 0 ? ? b a is called a linear equation in two variables x and y . General Form of a Pair of Linear Equation in Two Variables: A pair of linear equations in two variables x and y can be represented as follows: a 1 x + b 1 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 System of simultaneous linear equations: Consistent system: A system of simultaneous linear equations is said to be consistent if it has at least one solution. Inconsistent system: A system of simultaneous linear equations is said to be inconsistent if it has no solution. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES A pair of linear equations in two variables can be solved by the: (i) Graphical Method. (ii) Algebraic Method. Algebraic methods are of three types: (a) Substitution Method. (b) Elimination Method. (c) Cross-multiplication Method. Graphical Method: (i) Intersecting, if ? here, the equations have a unique solution, and pair of equations is said to be consistent. (ii) Parallel, if = ? here, the equations have no solution, and pair of equations is said to be inconsistent. (iii) Coincident, if = = here, the equations have infinitely many solutions, and pair of equations is said to be consistent. Algebraic Method: a) Substitution Method: Let us consider any two equations ) ( .......... 2 2 A b y a x = + ) ( .......... 4 B b y a x = - Page 3 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Algebraic Expression: A combination of constants and variables, connected by four fundamental arithmetic operations of +, -, x and ÷ is called an algebraic expression. Equation: An algebraic expression with equal to sign (=) is called an equation. Without an equal to sign, it is considered as an expression only. Linear Equation: If the greatest exponent of the variable(s) in an equation is one, then the equation is said to be a linear equation. If number of variable used in linear equation is one, then equation is said to be linear equation in two variables. Linear Equation in Two Variables: An equation which can be put in the form 0 = + + c by ax , where a, b and c are real numbers, and 0 , 0 ? ? b a is called a linear equation in two variables x and y . General Form of a Pair of Linear Equation in Two Variables: A pair of linear equations in two variables x and y can be represented as follows: a 1 x + b 1 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 System of simultaneous linear equations: Consistent system: A system of simultaneous linear equations is said to be consistent if it has at least one solution. Inconsistent system: A system of simultaneous linear equations is said to be inconsistent if it has no solution. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES A pair of linear equations in two variables can be solved by the: (i) Graphical Method. (ii) Algebraic Method. Algebraic methods are of three types: (a) Substitution Method. (b) Elimination Method. (c) Cross-multiplication Method. Graphical Method: (i) Intersecting, if ? here, the equations have a unique solution, and pair of equations is said to be consistent. (ii) Parallel, if = ? here, the equations have no solution, and pair of equations is said to be inconsistent. (iii) Coincident, if = = here, the equations have infinitely many solutions, and pair of equations is said to be consistent. Algebraic Method: a) Substitution Method: Let us consider any two equations ) ( .......... 2 2 A b y a x = + ) ( .......... 4 B b y a x = - PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Get the value of x in terms of y from equation (A), Substitute the value of x in equation (B) to get value of y. b) Elimination Method: Consider two equations ) ( .......... 2 2 A b y a x = + ) ( .......... 4 B b y a x = - Step1: Comparing the coefficients of x and y. ) .......( 2 2 i ab ay bx = + ) .......( 4 ii ab ay bx = - Step2: Simplify the equation either by adding or subtracting. Step 3: After simplifying you will find one value either â€˜xâ€™ or â€˜yâ€™. Step4: Substitute value of â€˜xâ€™ or â€˜yâ€™ in any of one equation to get the value of other variable. c) Cross Multiplication Method: Consider two equations: ) ( .......... 2 2 A b y a x = + ) ( .......... 4 B b y a x = - Step1: Below â€˜xâ€™ write the coefficients of â€˜yâ€™ and the constant terms. Below â€˜yâ€™ write the coefficients of â€˜xâ€™ and the constant terms. Below 1, write the coefficients of â€˜xâ€™ and â€˜yâ€™. Page 4 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Algebraic Expression: A combination of constants and variables, connected by four fundamental arithmetic operations of +, -, x and ÷ is called an algebraic expression. Equation: An algebraic expression with equal to sign (=) is called an equation. Without an equal to sign, it is considered as an expression only. Linear Equation: If the greatest exponent of the variable(s) in an equation is one, then the equation is said to be a linear equation. If number of variable used in linear equation is one, then equation is said to be linear equation in two variables. Linear Equation in Two Variables: An equation which can be put in the form 0 = + + c by ax , where a, b and c are real numbers, and 0 , 0 ? ? b a is called a linear equation in two variables x and y . General Form of a Pair of Linear Equation in Two Variables: A pair of linear equations in two variables x and y can be represented as follows: a 1 x + b 1 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 System of simultaneous linear equations: Consistent system: A system of simultaneous linear equations is said to be consistent if it has at least one solution. Inconsistent system: A system of simultaneous linear equations is said to be inconsistent if it has no solution. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES A pair of linear equations in two variables can be solved by the: (i) Graphical Method. (ii) Algebraic Method. Algebraic methods are of three types: (a) Substitution Method. (b) Elimination Method. (c) Cross-multiplication Method. Graphical Method: (i) Intersecting, if ? here, the equations have a unique solution, and pair of equations is said to be consistent. (ii) Parallel, if = ? here, the equations have no solution, and pair of equations is said to be inconsistent. (iii) Coincident, if = = here, the equations have infinitely many solutions, and pair of equations is said to be consistent. Algebraic Method: a) Substitution Method: Let us consider any two equations ) ( .......... 2 2 A b y a x = + ) ( .......... 4 B b y a x = - PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Get the value of x in terms of y from equation (A), Substitute the value of x in equation (B) to get value of y. b) Elimination Method: Consider two equations ) ( .......... 2 2 A b y a x = + ) ( .......... 4 B b y a x = - Step1: Comparing the coefficients of x and y. ) .......( 2 2 i ab ay bx = + ) .......( 4 ii ab ay bx = - Step2: Simplify the equation either by adding or subtracting. Step 3: After simplifying you will find one value either â€˜xâ€™ or â€˜yâ€™. Step4: Substitute value of â€˜xâ€™ or â€˜yâ€™ in any of one equation to get the value of other variable. c) Cross Multiplication Method: Consider two equations: ) ( .......... 2 2 A b y a x = + ) ( .......... 4 B b y a x = - Step1: Below â€˜xâ€™ write the coefficients of â€˜yâ€™ and the constant terms. Below â€˜yâ€™ write the coefficients of â€˜xâ€™ and the constant terms. Below 1, write the coefficients of â€˜xâ€™ and â€˜yâ€™. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Step2: Simplify the equations. Step3: After simplifying we get value of â€˜xâ€™ and â€˜yâ€™. Equation Reducible to a There are several situations which can be mathematically equations that are not linear to start with. But we a of linear equations. We will be easily able to understand this concept of equation reducible to a pair of linear equation in two variables.. We shall discuss the solution of such pairs of equations which are not linear but can be reduced to linear form by making some suitable substitutions. We now explain this process through some examples. Example: Solve the pair of equations: Solution: Let us write the given pair of equations as PAIR OF LINEAR EQUATIONS IN TWO VARIABLES the equations. After simplifying equate the values of â€˜xâ€™ and â€˜yâ€™ with constant terms, get value of â€˜xâ€™ and â€˜yâ€™. a Pair of Linear Equations In Two Variables There are several situations which can be mathematically represented by two equations that are not linear to start with. But we alter them so as to We will be easily able to understand this concept of equation reducible to a pair of linear equation in two variables.. shall discuss the solution of such pairs of equations which are not linear but can be reduced to linear form by making some suitable substitutions. We now explain this process through some examples. the pair of equations: us write the given pair of equations as PAIR OF LINEAR EQUATIONS IN TWO VARIABLES and â€˜yâ€™ with constant terms, f Linear Equations In Two Variables represented by two lter them so as to reduce to pair We will be easily able to understand this concept of equation reducible to a pair of shall discuss the solution of such pairs of equations which are not linear but can be reduced to linear form by making some suitable substitutions. We now Page 5 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Algebraic Expression: A combination of constants and variables, connected by four fundamental arithmetic operations of +, -, x and ÷ is called an algebraic expression. Equation: An algebraic expression with equal to sign (=) is called an equation. Without an equal to sign, it is considered as an expression only. Linear Equation: If the greatest exponent of the variable(s) in an equation is one, then the equation is said to be a linear equation. If number of variable used in linear equation is one, then equation is said to be linear equation in two variables. Linear Equation in Two Variables: An equation which can be put in the form 0 = + + c by ax , where a, b and c are real numbers, and 0 , 0 ? ? b a is called a linear equation in two variables x and y . General Form of a Pair of Linear Equation in Two Variables: A pair of linear equations in two variables x and y can be represented as follows: a 1 x + b 1 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 System of simultaneous linear equations: Consistent system: A system of simultaneous linear equations is said to be consistent if it has at least one solution. Inconsistent system: A system of simultaneous linear equations is said to be inconsistent if it has no solution. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES A pair of linear equations in two variables can be solved by the: (i) Graphical Method. (ii) Algebraic Method. Algebraic methods are of three types: (a) Substitution Method. (b) Elimination Method. (c) Cross-multiplication Method. Graphical Method: (i) Intersecting, if ? here, the equations have a unique solution, and pair of equations is said to be consistent. (ii) Parallel, if = ? here, the equations have no solution, and pair of equations is said to be inconsistent. (iii) Coincident, if = = here, the equations have infinitely many solutions, and pair of equations is said to be consistent. Algebraic Method: a) Substitution Method: Let us consider any two equations ) ( .......... 2 2 A b y a x = + ) ( .......... 4 B b y a x = - PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Get the value of x in terms of y from equation (A), Substitute the value of x in equation (B) to get value of y. b) Elimination Method: Consider two equations ) ( .......... 2 2 A b y a x = + ) ( .......... 4 B b y a x = - Step1: Comparing the coefficients of x and y. ) .......( 2 2 i ab ay bx = + ) .......( 4 ii ab ay bx = - Step2: Simplify the equation either by adding or subtracting. Step 3: After simplifying you will find one value either â€˜xâ€™ or â€˜yâ€™. Step4: Substitute value of â€˜xâ€™ or â€˜yâ€™ in any of one equation to get the value of other variable. c) Cross Multiplication Method: Consider two equations: ) ( .......... 2 2 A b y a x = + ) ( .......... 4 B b y a x = - Step1: Below â€˜xâ€™ write the coefficients of â€˜yâ€™ and the constant terms. Below â€˜yâ€™ write the coefficients of â€˜xâ€™ and the constant terms. Below 1, write the coefficients of â€˜xâ€™ and â€˜yâ€™. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES Step2: Simplify the equations. Step3: After simplifying we get value of â€˜xâ€™ and â€˜yâ€™. Equation Reducible to a There are several situations which can be mathematically equations that are not linear to start with. But we a of linear equations. We will be easily able to understand this concept of equation reducible to a pair of linear equation in two variables.. We shall discuss the solution of such pairs of equations which are not linear but can be reduced to linear form by making some suitable substitutions. We now explain this process through some examples. Example: Solve the pair of equations: Solution: Let us write the given pair of equations as PAIR OF LINEAR EQUATIONS IN TWO VARIABLES the equations. After simplifying equate the values of â€˜xâ€™ and â€˜yâ€™ with constant terms, get value of â€˜xâ€™ and â€˜yâ€™. a Pair of Linear Equations In Two Variables There are several situations which can be mathematically represented by two equations that are not linear to start with. But we alter them so as to We will be easily able to understand this concept of equation reducible to a pair of linear equation in two variables.. shall discuss the solution of such pairs of equations which are not linear but can be reduced to linear form by making some suitable substitutions. We now explain this process through some examples. the pair of equations: us write the given pair of equations as PAIR OF LINEAR EQUATIONS IN TWO VARIABLES and â€˜yâ€™ with constant terms, f Linear Equations In Two Variables represented by two lter them so as to reduce to pair We will be easily able to understand this concept of equation reducible to a pair of shall discuss the solution of such pairs of equations which are not linear but can be reduced to linear form by making some suitable substitutions. We now PAIR OF LINEAR EQUATIONS IN TWO VARIABLES These equations are not in the form ax + by + c = 0. However, if we Substitute = p and = q in Equations (1) and (2) we get, 2p + 3q = 13 --------------------------------------- (3) 5p - 4q = -2 ---------------------------------------- (4) So, we have expressed the equations as a pair of linear equations. Now, you can use any method to solve these equations, and get p = 2 and q = 3. We know that p = and q = Substitute the values of p and q to get = 2 i.e. x = and = 3 i.e. y = Verification: By substituting x = and y = in the given equations, we find that both the equations are satisfied.Read More

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