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PPT on Linear Equations in two variables
The document explains linear equations in two variables, describing their form and methods for solving them, including graphical and algebraic approaches. It outlines different types of solution scenarios—one solution, no solution, and infinitely many solutions—and details three algebraic methods: substitution, elimination, and cross-multiplication. Examples illustrate the application of these methods to find solutions for given sets of equations.
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PPT on Linear Equations in two variables
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- 2. Linear Equations Definition ofa Linear Equation A linear equation in two variable x is an equation that can be written in the form ax + by + c = 0, where a ,b and c are real numbers and a and b is not equal to 0. An example of a linear equation in x is 2x – 3y + 4 = 0.
- 3. A pair oflinear equations in two variables can be solved by the: (i) Graphically method (ii) Algebraic method
- 4. GRAPHICAL SOLUTIONS OFA LINEAR EQUATION Let us consider the following system of two simultaneous linear equations in two variable. 2x – y = -1 3x + 2y = 9 Here we assign any value to one of the two variables and then determine the value of the other variable from the given equation.
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- 7. Types of Solutionsof Systems of Equations • One solution – the lines cross at one point • No solution – the lines do not cross • Infinitely many solutions – the lines coincide
- 8. Algebraic method TO SOLVEA PAIR OF LINEAR EQUATION IN TWO VARIABLE
- 9. To solve apair of linear equations in two variables algebraically, we have following methods:(i) Substitution method (ii) Elimination method (iii) Cross-multiplication method
- 10. SUBSTITUTION METHOD STEPS Obtain thetwo equations. Let the equations be a1x + b1y + c1 = 0 ----------- (i) a2x + b2y + c2 = 0 ----------- (ii) Choose either of the two equations, say (i) and find the value of one variable , say ‘y’ in terms of x Substitute the value of y, obtained in the previous step in equation (ii) to get an equation in x
- 11. Solve the equationobtained in the previous step to get the value of x. Substitute the value of x and get the value of y. Let us take an example x + 2y = -1 ------------------ (i) 2x – 3y = 12 -----------------(ii)
- 12. x + 2y= -1 x = -2y -1 ------- (iii) Substituting the value of x in equation (ii), we get 2x – 3y = 12 2 ( -2y – 1) – 3y = 12 - 4y – 2 – 3y = 12 - 7y = 14 ; y = -2 ,
- 13. Putting the valueof y in eq. (iii), we get x = - 2y -1 x = - 2 x (-2) – 1 =4–1 =3 Hence the solution of the equation is ( 3, - 2 )
- 14. ELIMINATION METHOD • Inthis method, we eliminate one of the two variables to obtain an equation in one variable which can easily be solved. Putting the value of this variable in any of the given equations, the value of the other variable can be obtained. • For example: we want to solve, 3x + 2y = 11 2x + 3y = 4
- 15. Let 3x +2y = 11 --------- (i) 2x + 3y = 4 ---------(ii) Multiply 3 in equation (i) and 2 in equation (ii) and subtracting eq iv from iii, we get 9x + 6y = 33 ------ (iii) 4x + 6y = 8 ------- (iv) 5x = 25 x=5
- 16. • putting thevalue 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
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