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Presentation On Linear Equation In 2 Variable

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Published in: Mathematics
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Solving of linear equation

Pratibha T / Dubai

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  1. Linear equation in 2 variables Prepared by : Pratibha Thakur Ranjan
  2. Linear Equation An equation is said to be linear equation in two variables if it is written in the form of ax + by + c=O, where a, b & c are real numbers and the coefficients of x and y, i.e a and b respectively, are not equal to zero. For example 2x+3y+7=O where a=2, b=3, c=7 which is not equal to zero. There are 2 methods for solving these Linear equation Graphical Algebraic
  3. Algebraic Method Cross multiplication Elimination Substitutions
  4. Substitution Method 4,4ß Find Y in terms of X from one equation. Substitute the value of Y in the other equation. Solve for X. Substitute the value of X to get the value of Y. Can do vice versa also. For example, to solve the system of equations 3x-y-8=O --(ii) Use (ii), to get the value of y, y=3x-8 -—-(iii) Now use this yin equation (i) 3x+2(3x-8)-2=O Open the bracket by multiplying 2 3X+6x-16-2=o 9x-18=o 9x=18 x=18/9 , Add x term with x & constant term Shift 18 to other side to get value of x. Now use the value of x in equation (iii) to get y. 6-8=-2 -2 Desired solution (x, y) = (2, -2)
  5. Elimination Method Multiply the given equation by some suitable constant to make coefficient of one variable numerically equal. Add or subtract accordingly to eliminate one variable of same coefficient. Solve the equation and get value. To find another variable put the value obtained in one of the equation. Consider the following example 3x-y = 8 - Now here in eq.(i) the coefficient of y is 2 and in eq. (ii) the coefficient of y is -1. So here I will multiply eq.(ii) with 2 to make coefficient same as eq.(i) 2(3x-y = 8) = 16 (iii) , Since signs are opposite so la going to add eq.(i) & eq,(iii). 3x+ +6x- y; 16 18 18/9, 2 y get eliminated now solve for x. Use in any equation to get value of y. 3(2)- y=8, -2 Desired solution (x, y) = (2, -2)
  6. Cross Multiplication: This is the simplest method and gives the accurate value of the variables. Cross multiplication is only applicable when we have a pair of linear equations in two variables . Assume two linear equation be al x + bly+ cl = O, and & a2x +b2y + c2 = 0. We can get ratio like this - Cia: — alb: — Now we can get value of x and y like this Use this table bicz-bzcl /albz-azbl and yzc1a2-c2a1 /ct1>2-a2b1 .
  7. Lets see an example 3x-y-8 - Here 3, biz 2, -2 , 3, -1 and -8 Now use formula to find x & v. -16-2/-3-6 = -18/-9 = 2 Y = = -6+24/-9 = 18/-9 = -2 and y Solution is (x, y) = (2, -2)
  8. Graphical Method A linear equation in two variables when plotted on a graph defines a line. So, this means when a pair of linear equations is plotted, two lines are defined. Now, there are two lines in a plane These lines can: •intersect each other, •be parallel to each other, or •coincide with each other. The point(s) where the two lines intersect will give the solutions of the pair of linear equations, graphically. Condition 1: Intersecting Lines If al/a2 bl/b2, then the pair of linear equations alx + bly + cl = O, a2x + b2y + c2 = O has a unique solution & consistent. Condition 2: Coincident Lines If al/a2 = bl/b2 = cl/c2, then the pair of linear equations alx + bly + = O, a2X+ b2y + cz = O has infinite solutions. A pair of linear equations, which has unique or infinite solutions is said to be a consistent or dependent. Condition 3: Parallel Lines If al/a2 = bl/b2 cl/c2 , then a pair of linear equations alx + bly + = O, a2X + b2y + cz = O has no solution. A pair of linear equations which has no solution is said to be an inconsistent pair of linear equations
  9. For an example & 3x+2y=2 3x-y=8 First, solve each equation for "y z" Or change each equation in y = mx + b form. From 1st eq. we got 3x-8 and from 2nd we got -3x+2/2 Now create table for both of the equations -3 -2 -1 1 2 3 3x-8 14 11 -3 -2 -1 1 2 3 -3x+2/2 = 11/2 = 5/2 1/2 = -7/2 Here in the graph we can clearly see intersection point at (2, -2). Since the graph is intersecting so it have unique sol.
  10. 4,4ß We can solve in other way around & 3x+2y=2 3x-y=8 As here from equation 1 we get al: 3 , bl n rom equation , we get a Now lets check the ratio al/a2 = 3/3 1 bl/b2 = -1/2 & cl/c2 = -8/-2 4 Here al/a2 bl/b2 So it have unique solution & line are intersecting with each other which it is clearly evident in the above graph.
  11. Thank you

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