This document discusses polynomials and systems of linear equations. It defines polynomials as expressions involving variables and coefficients using only addition, subtraction, multiplication, and division. Polynomials are used in many areas of mathematics and science. The document also defines linear equations and discusses different methods for solving systems of linear equations graphically or algebraically, including substitution, elimination, and cross-multiplication methods. It provides examples to illustrate each method.

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

12. CROSS MULTIPLICATION
METHOD

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