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Presentation On Linear Equations 1

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Pair of Linear Equations

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  1. 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 apeb1Y+c1=O a2x+b2Y+c2=O Where 'a', 'b' and 'c' are not equal to real numbers 'a' and 'b' are not equal to zero.
  2. It is true for any linear equation that each solution (x, y) of a linear equation in two variables, ax + by + c = O, corresponds to a point on the line representing the equation, and vice versa.
  3. Three possible solutions for a given pair of linear equations:
  4. a. The two lines will intersect at one point. (Intersecting lines) a. The two line will not intersect, i.e, they are parallel. (Parallel lines) a. The two lines will be coincident. (Coincident lines over lap)
  5. (b)
  6. DERIVING THE SOLUTION THROUGH GRAPHICAL METHOD Let us consider the following system of two simultaneous linear equations in two variable. 2x-y=-1 We can determine the value of the a variable by substituting any value for the other variable, as done in the given examples 2x-y=-1 2y=9-3x 5 6
  7. NO ? LVIOO ? 9 9 6 = + Xt ? NO ? LV()O ? 0
  8. Consistent and Inconsistent pair of Linear equations (i) the lines may intersect in a single point. In this case, the pair of equations has a unique solution (consistent pair of equations). (ii) the lines may be parallel. In this case, the equations have no solution (inconsistent pair of equations). (iii) the lines may be coincident. In this case, the equations have infinitely many solutions [dependent (consistent) pair of equations].
  9. Comparison of Ratios
  10. al Pair of lines 1 3 3x+4y-20=O -2 4 3 6 4 Table 3.4 Compare the ratios _9 Graphical Algebrai representation interpret Intersecting Exactly 2x+3y-9=O 4x+6y—1S=0 x+2y—4=0 2x+4y-12=O 2 4 1 2 lines Coincident lines Parallel lines solution (unique) Infinitely ny sol o solut
  11. 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
  12. Substituting the value of x inequation (ii), we get Putting the value of y in eq. (iii), we get x + 2y = -1 - 2x-3y = 12 x + 2y = -1 - ------- (iii) -3y = 12 7y = 14 - l) — 3y = 12- = 12- = 12 - 14 = 7y Hence the solution of the equation is 3
  13. 3.4.2 Elimination Method Now let us consider another method of eliminating (i.e., removing) one variable. This is sometimes more convenient than the substitution method. Let us see how this method works. Step 1 : First multiply both the equations by some suitable non-zero constants to make the coefficients of one variable (either x or y) numerically equal. Step 2 : Then add or subtract one equation from the other so that one variable gets eliminated. If you get an equation in one variable, go to Step 3. If in Step 2, we obtain a true statement involving no variable, then the original pair of equations has infinitely many solutions. If in Step 2, we obtain a false statement involving no variable, then the original pair of equations has no solution, i.e., it is inconsistent. Step 3 : Solve the equation in one variable (x or y) so obtained to get its value. Step 4 : Substitute this value of x (or y) in either of the original equations to get the value of the other variable. Now to illustrate it, we shall solve few more examples.
  14. + 3y ---------(iii) - 3y = 33 (iii) — (iv) Putting the value of x in equation (ii) we get, => Hence, x = 5 and = 8---------(ii) -----------(iii) + 33 = ------------(iv) = 25 + 3y = 4 2 x 5 + 3y = 4 10 + 3y = 4 3y = 4— 10 3y=-6 y=-2
  15. DERIVING THE SOLUTION THROUGH ROSS-MULTIPLICATION METHO 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 axl + byl + Cl = O; ax2 + bY2 + c2 = O blC2 —b2C1 x= alb2 —a2b1 Cla2 —c2a1 alb2 —a2b1
  16. 1 x Blc2-b2c1 blC2 -b2Cl x: alb2 —a2b1 cla2—c2a1 alb2 —a2b1 Cla2 —c2a1 alb2 —a2b1
  17. Example: 8? + 5? 9 ? ? -20-(-18) 4 ? -2 5 ????—???? -27-(-32) ? -2 and 1 ??2 —??? 16-15 5
  18. EQUATIONS REDUCIBLE TO PAIR OF LINEAR EQUATION IN TWO VARIABLES In case of equations which are not linear, like - 13 We can turn the equations into linear equations by substituting
  19. 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
  20. ????? ???

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