Fundamentals of Polynomials
*Equation :
A statement of equality involving one or more variables is called an equation (or) An equation is a statement in which two algebraic expressions are equal.
Ex : (i) 2x-4=6 (ii) x+2= 3y-4
*Linear equation :
An equation involving one variable with highest power 1, is called a linear equation in that variable.
Ex : (i) 2x+5 =7 (ii) 4y = 2
*Solution of a linear equation : When the value of the variable satisfies the given equation then that value is called the solution (root) of the given equation.
*Rules for solving a Linear Equation :
Solving a linear equation involves isolating the variable on one side of the equation. Here are the general rules and steps to solve a linear equation:
Understand the Equation:
Identify the variable(s) in the equation. Typically, a linear equation is in the form:
ax+b=c, where a,b and c are constants, and x is the variable.
Combine Like Terms:
If there are like terms on either side of the equation, combine them. Like terms have the same variable and exponent.
Isolate the Variable:
Perform operations to isolate the variable on one side of the equation. Use inverse operations to undo the operations present in the equation. Here are common operations and their inverses:
Addition and Subtraction:
Undo addition with subtraction and vice versa.
Multiplication and Division:
Undo multiplication with division and vice versa.
Perform Operations Symmetrically:
Perform the same operation on both sides of the equation to maintain balance. The goal is to simplify the equation and isolate the variable.
Check Your Solution:
After solving for the variable, substitute the solution back into the original equation to ensure that both sides are equal. If the equation holds true, then the solution is correct.
No Solution or Infinite Solutions:
In some cases, a linear equation may have no solution or infinite solutions. This can happen when simplifying the equation leads to a statement that is always true (e.g., 0=0) or always false (e.g., 3=5)
Example
Solve the equation 2x-5=11
Add 5 to both sides: 2x =16.
Divide by 2:x=8.
Check the solution: 2(8)-5=16-5=11
The equation holds true, so x=8 is the solution.
The equality of a linear equation is not changed, when
(1) the same number is added to both sides of the equation,
(2) the same number is subtracted from both sides of the equation,
(3) both sides of the equation are multiplied by the same non-zero number,
(4) both sides of the equation are divided by the same non-zero number.