Ultimate guide to systems of equations

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1. Includes sample problems Ultimate Guide to solving systems of equations

2. Table of Contents: Introduction Solving Systems by Graphing Solving Systems by Substitution Solving Systems by Elimination Special Cases Summary

3. Introduction to Systems of Equations A pair of linear equations is said to form a system of linear equations in the standard form a 1 x+b 1 y+c 1 =0 a 2 x+b 2 y+c 2 =0 Where ‘a’ , ‘b’ and ‘c’ are not equal to real numbers ‘a’ and ‘b’ are not equal to zero.

4. Obtaining the solution By Graphing 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 2 Y 1 5 X 3 -1 Y 6 X=(y-1)/2 y=2x+1 2y=9-3x x=(9-2y)/3 2x – y = -1 3x + 2y = 9

5. 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1 1 2 3 4 5 -1 -2 -3 -4 -5 ( 1,3 ) ( 2,5 ) (- 1,6 ) ( 0,1 ) x=1 X 2 y 1 5 X 3 -1 Y 6 EQUATION 1 EQUATION 2 ‘X’ intercept = ‘Y’ intercept = 2x – y = -1 3x + 2y = 9 y =3

6. ax 1 + by 1 + c 1 = 0; ax 2 + by 2 + c 2 = 0 a 1 b 1 c 1 c 2 a 2 b 2 = i) ii) iii) = a 1 b 1 a 2 b 2 = a 1 b 1 c 1 c 2 a 2 b 2 = = Intervening Lines; Infinite Solutions Intersecting Lines; Definite Solution Parallel Lines; No Solution

7. 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. For example x + 2y = - 1 ; 2x – 3y = 12

8. 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 The solution of the equation is ( 3, - 2 )

9. 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

10. 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 4 x + 6y = 8---------(ii) x2 5x = 25 x = 5 P utting 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

11. Deriving the solution through the 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 ax 1 + by 1 + c 1 = 0; ax 2 + by 2 + c 2 = 0 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 X= Y=

12. X B 1 c 2 -b 2 c 1 Y c 1 a 2 –c 2 a 1 = 1 a 1 b 2 –a 2 b 1 = 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 X= Y=

13. 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 B 1 c 2 -b 2 c 1 Y c 1 a 2 –c 2 a 1 = 1 a 1 b 2 –a 2 b 1 =

14. Equations reducible to A pair of linear equations 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 =

15. 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 = &

16. 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