Pair of linear equations in two variable
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The document provides an overview of solving pairs of linear equations in two variables, including methods such as graphical, substitution, elimination, and cross-multiplication. It illustrates examples for each method to derive solutions and emphasizes the ability to transform non-linear equations into linear equations for resolution. Additionally, it summarizes key insights regarding simultaneous linear equations and their solutions.
Pair of linear equations in two variable
- 1. PAIR OF LINEAR EQUATIONS IN TWO VARIABLE
- 2. INTRODUCTION TO PAIR OF LINEAR EQUATION IN TWO VARIABLES A pair of linear equation is said to form a system of simultaneous linear equation in the standard form a1x+b1y+c1=0 a2x+b2y+c2=0 Where ‘a’, ‘b’ and ‘c’ are not equal to real numbers ‘a’ and ‘b’ are not equal to zero.
- 3. DERIVING THE SOLUTION THROUGH GRAPHICAL METHOD Let us consider the following system of two simultaneous linear equations in two variable. 2x – y = -1 ;3x + 2y = 9 We can determine the value of the a variable by substituting any value for the other variable, as done in the given examples X 0 2 Y 1 5 X 3 -1 Y 0 6 X=(y-1)/2 y=2x+1 2y=9-3x x=(9-2y)/3 2x – y = -1 3x + 2y = 9
- 4. 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 -2 -3 -4 -5 (1,3) (2,5) (-1,6) (0,1) X=1 Y=3 X 0 2 Y 1 5 X 3 -1 Y 0 6 EQUATION 1 EQUATION 2 ‘X’ intercept = 1 ‘Y’ intercept = 3 2x – y = -1 3x + 2y = 9
- 5. ax1 + by1 + c1 = 0; ax2 + by2 + c2 = 0 a1 b1 c1 c2a2 b2 =i) ii) iii) = a1 b1 a2 b2 = a1 b1 c1 c2a2 b2 == Intervening Lines; Infinite Solutions Intersecting Lines; Definite Solution Parallel Lines; No Solution
- 6. DERIVING THE SOLUTION THROUGH SUBSTITUTION METHOD This method involves substituting the value of one variable, say x , in terms of the other in the equation to turn the expression into a Linear Equation in one variable, in order to derive the solution of the equation . For example x + 2y = -1 ;2x – 3y = 12
- 7. 2x – 3y = 12 ----------(ii) x = -2y -1 x = -2 x (-2) – 1 = 4–1 x = 3 x + 2y = -1 -------- (i) x + 2y = -1 x = -2y -1 ------- (iii) Substituting the value of x inequation (ii), we get 2x – 3y = 12 2 ( -2y – 1) – 3y = 12 - 4y – 2 – 3y = 12 - 7y = 14 = 12 - 14 = 7y y = -2 Putting the value of y in eq. (iii), we get Hence the solution of the equation is ( 3, - 2 )
- 8. DERIVING THE SOLUTION THROUGH ELIMINATION METHOD In this method, we eliminate one of the two variables to obtain an equation in one variable which can easily be solved. The value of the other variable can be obtained by putting the value of this variable in any of the given equations. For example: 3x + 2y = 11 ;2x + 3y = 4
- 9. 3x + 2y = 11 --------- (i) 2x + 3y = 4 ---------(ii) 3x + 2y = 11 x3- 9x - 3y = 33---------(iii) =>9x + 6y = 33-----------(iii) 4x + 6y = 8------------(iv) (-) (-) (-) (iii) – (iv) => x3 2x + 3y = 4 4x + 6y = 8---------(ii) x2 5x = 25 x = 5 Putting the value of x in equation (ii) we get, => 2x + 3y = 4 2 x 5 + 3y = 4 10 + 3y = 4 3y = 4 – 10 3y = - 6 y=-2 Hence, x = 5 and y = -2
- 10. DERIVING THE SOLUTION THROUGH CROSS-MULTIPLICATION METHOD 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 ax1 + by1 + c1 = 0; ax2 + by2 + c2 = 0 b1c2 –b2c1 a1b2 –a2b1 c1a2 –c2a1 a1b2 –a2b1 X= Y=
- 11. X B1c2-b2c1 Y c1a2 –c2a1 = 1 a1b2 –a2b1 = b1c2 –b2c1 a1b2 –a2b1 c1a2 –c2a1 a1b2 –a2b1 X= Y=
- 12. Example: 8x + 5y – 9 = 0 3x + 2y – 4 = 0 X -20-(-18) Y -27-(-32) = 1 16-15 = X Y 1 1-2 5 = X -2 Y 5 =1 1 X = -2 and Y = 5 X B1c2-b2c1 Y c1a2 –c2a1 = 1 a1b2 –a2b1 =
- 13. EQUATIONS REDUCIBLE TO PAIR OF LINEAR EQUATION IN TWO VARIABLES In case of equations which are not linear, like We can turn the equations into linear equations by substituting 2 3 13 x y = 5 4 -2 x y =+ - 1 p x = 1 q y =
- 14. The resulting equations are 2p + 3q = 13 ; 5p - 4q = -2 These equations can now be solved by any of the aforementioned methods to derive the value of ‘p’ and ‘q’. ‘p’ = 2 ;‘q’ = 3 We know that 1 p x = 1 q y = 1 X 2 = 1 Y 3 = &
- 15. SUMMARY • Insight to Pair of Linear Equations in Two Variable • Deriving the value of the variable through • Graphical Method • Substitution Method • Elimination Method • Cross-Multiplication Method • Reducing Complex Situation to a Pair of Linear Equations to derive their solution