Mitrocks
- 1. Created By MIT SHAH Class-10 C Roll No-44
- 2. INTRODUCTION IN MATHEMATICS,A POLYNOMIAL IS AN EXPRESSION CONSISTING OF VARIABLES AND COEFFICIENTS THAT INVOLVES ONLY THE OPERATIONS OF ADDITION,SUBTRACTION,MULTIPLICATION AND DIVISION. POLYNOMIALS APPEAR IN A WIDE VARIETY OF AREAS OF MATHEMATICS AND SCIENCE,FOR EXAMPLE-THEY ARE USED TO FORM POLYNOMIALS EQUATIONS,WHICH ANCODE A WIDE RANGE OF PROBLEMS,FROM ELEMENTARY WORD PROBLEMS TO COMPLICATED PROBLEMS IN RHE SCIENCES,THEY ARE USED TO DEFINE POLYNOMIAL FUNCTIONS,WHICH APPEAR IN SETTINGS RANGING FROM BASIC CHEMISTRY AND PHYSICS TO ECONOMICS AND SOCIAL SCIENCE,THEY ARE USED IN CALCULUS AND NUMERICAL ANALYSIS TO APPROXIMATE OTHER FUNCTIONS. IN ADVANCED MATHEMATICS,POLYNOMIALS ARE USED TO CONSTRUCT POLYNOMIAL RINGS AND ALGEBRAIC VARIETIES.
- 3. Linear Equations Definition of a 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.
- 4. GRAPHICAL SOLUTIONS OF A 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 Solutions of 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. TO SOLVE A PAIR OF LINEAR EQUATION IN TWO VARIABLES WE HAVE FOLL METHODS- <1>SUBSTITUTION METHOD <2.ELIMINATION METHOD <3>CROSS-MULTIPLICATION <1>SUBSTITUTION METHOD STEPS Obtain the two 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
- 9. Solve the equation obtained 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) 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 Putting the value of y in eq. (iii), we get x = - 2y -1 x = - 2 x (-2) – 1 = 4 – 1 =3 H ence the solution of the equation is ( 3, - 2 )
- 10. ELIMINATION METHOD • In this 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
- 11. 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 •putting 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
- 13. CROSS MULTIPLICATION METHOD Let’s consider the general form of a pair of linear equations. To solve this pair of equations for 푥 and 푦 using cross-multiplication, we’ll arrange the variables and their coefficients , and , and the constants and We can convert non linear equations in to linear equation by a suitable substitution