Linear equation in two variables

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Transcript of Linear equation in two variables

  • 1. PRESENTATION ONPRESENTATION ON LINEAR EQUATION INLINEAR EQUATION IN TWO VARIABLESTWO VARIABLES Submitted to:- Submitted by:-

2. System of equations orSystem of equations or simultaneous equations simultaneous equations A pair of linear equations in twoA pair of linear equations in two variables is said to form a system ofvariables is said to form a system of simultaneous linear equations.simultaneous linear equations. For Example, 2x 3y + 4 = 0For Example, 2x 3y + 4 = 0 x + 7y 1 = 0x + 7y 1 = 0 Form a system of two linear equationsForm a system of two linear equations in variables x and y.in variables x and y. 3. The general form of a linear equation inThe general form of a linear equation in two variables x and y istwo variables x and y is ax + by + c = 0 , a =/= 0, b=/=0, whereax + by + c = 0 , a =/= 0, b=/=0, where a, b and c being real numbers.a, b and c being real numbers. A solution of such an equation is a pair ofA solution of such an equation is a pair of values, one for x and the other for y, whichvalues, one for x and the other for y, which makes two sides of the equation equal.makes two sides of the equation equal. Every linear equation in two variables hasEvery linear equation in two variables has infinitely many solutions which can beinfinitely many solutions which can be represented on a certain line.represented on a certain line. 4. GRAPHICAL SOLUTIONS OF AGRAPHICAL SOLUTIONS OF A LINEAR EQUATIONLINEAR EQUATION Let us consider the following system ofLet us consider the following system of two simultaneous linear equations in twotwo simultaneous linear equations in two variable.variable. 2x y = -12x y = -1 3x + 2y = 93x + 2y = 9 Here we assign any value to one of the twoHere we assign any value to one of the two variables and then determine the value ofvariables and then determine the value of the other variable from the given equation.the other variable from the given equation. 5. Cartesian PlaneCartesian Plane x- axis y-axis Quadrant I (+,+) Quadrant II ( - ,+) Quadrant IV (+, - ) Quadrant III ( - , - ) origin 6. For the equationFor the equation 2x y = -1 ---(1)2x y = -1 ---(1) 2x +1 = y2x +1 = y Y = 2x + 1Y = 2x + 1 3x + 2y = 9 --- (2)3x + 2y = 9 --- (2) 2y = 9 3x2y = 9 3x 9- 3x9- 3x Y = -------Y = ------- 22 X 0 2 Y 1 5 X 3 -1 Y 0 6 7. XX Y Y (2,5) (-1,6) (0,3)(0,1) X= 1 Y=3 8. ALGEBRAIC METHODS OFALGEBRAIC METHODS OF SOLVING SIMULTANEOUSSOLVING SIMULTANEOUS LINEAR EQUATIONSLINEAR EQUATIONS The most commonly used algebraicThe most commonly used algebraic methods of solving simultaneous linearmethods of solving simultaneous linear equations in two variables areequations in two variables are Method of elimination by substitutionMethod of elimination by substitution Method of elimination by equating theMethod of elimination by equating the coefficientcoefficient Method of Cross- multiplicationMethod of Cross- multiplication 9. ELIMINATION BY SUBSTITUTIONELIMINATION BY SUBSTITUTION STEPSSTEPS Obtain the two equations. Let the equations beObtain the two equations. Let the equations be aa11x + bx + b11y + cy + c11 = 0 ----------- (i)= 0 ----------- (i) aa22x + bx + b22y + cy + c22 = 0 ----------- (ii)= 0 ----------- (ii) Choose either of the two equations, say (i) andChoose either of the two equations, say (i) and find the value of one variable , say y in termsfind the value of one variable , say y in terms of xof x Substitute the value of y, obtained in theSubstitute the value of y, obtained in the previous step in equation (ii) to get an equationprevious step in equation (ii) to get an equation in xin x 10. ELIMINATION BY SUBSTITUTIONELIMINATION BY SUBSTITUTION Solve the equation obtained in theSolve the equation obtained in the previous step to get the value of x.previous step to get the value of x. Substitute the value of x and get theSubstitute the value of x and get the value of y.value of y. Let us take an exampleLet us take an example x + 2y = -1 ------------------ (i)x + 2y = -1 ------------------ (i) 2x 3y = 12 -----------------(ii)2x 3y = 12 -----------------(ii) 11. SUBSTITUTION METHODSUBSTITUTION METHOD x + 2y = -1x + 2y = -1 x = -2y -1 ------- (iii)x = -2y -1 ------- (iii) Substituting the value of x inSubstituting the value of x in equation (ii), we getequation (ii), we get 2x 3y = 122x 3y = 12 2 ( -2y 1) 3y = 122 ( -2y 1) 3y = 12 - 4y 2 3y = 12- 4y 2 3y = 12 - 7y = 14 , y = -2 ,- 7y = 14 , y = -2 , 12. SUBSTITUTION METHODSUBSTITUTION METHOD Putting the value of y in eq (iii), we getPutting the value of y in eq (iii), we get x = - 2y -1x = - 2y -1 x = - 2 x (-2) 1x = - 2 x (-2) 1 = 4 1= 4 1 = 3= 3 Hence the solution of the equation isHence the solution of the equation is ( 3, - 2 )( 3, - 2 ) 13. ELIMINATION METHODELIMINATION METHOD In this method, we eliminate one of theIn this method, we eliminate one of the two variables to obtain an equation in onetwo variables to obtain an equation in one variable which can easily be solved.variable which can easily be solved. Putting the value of this variable in any ofPutting the value of this variable in any of the given equations, the value of the otherthe given equations, the value of the other variable can be obtained.variable can be obtained. For example: we want to solve,For example: we want to solve, 3x + 2y = 113x + 2y = 11 2x + 3y = 42x + 3y = 4 14. Let 3x + 2y = 11 --------- (i)Let 3x + 2y = 11 --------- (i) 2x + 3y = 4 ---------(ii)2x + 3y = 4 ---------(ii) Multiply 3 in equation (i) and 2 in equation (ii) andMultiply 3 in equation (i) and 2 in equation (ii) and subtracting eq iv from iii, we getsubtracting eq iv from iii, we get 9x + 6y = 33 ------ (iii)9x + 6y = 33 ------ (iii) 4x + 6y = 8 ------- (iv)4x + 6y = 8 ------- (iv) 5x = 255x = 25 => x = 5=> x = 5 15. putting the value of y in equation (ii) we get,putting the value of y in equation (ii) we get, 2x + 3y = 42x + 3y = 4 2 x 5 + 3y = 42 x 5 + 3y = 4 10 + 3y = 410 + 3y = 4 3y = 4 103y = 4 10 3y = - 63y = - 6 y = - 2y = - 2 Hence, x = 5 and y = -2Hence, x = 5 and y = -2