CLASS-7
CONCEPT OF EQUATION

CONCEPT OF EQUATION -

An equation is a mathematical statement that asserts the equality of two expressions. It is a fundamental concept in mathematics, used to express relationships between variables and constants.

Basic Structure of an Equation:-

  • An equation typically has two sides:-   a left-hand side (LHS) and a right-hand side (RHS), separated by an equals sign "=".
  • The general form of an equation is:-   LHS = RHS
  • Each side of the equation contains mathematical expressions involving numbers, variables, or a combination of both.

Examples of Equations:-

    1. Simple Equation:-

         x + 2 = 5

        Here, x is a variable, and the equation asserts that when you add 2 to x, the result is 5.

    2. Quadratic Equation:-

        x²− 4x + 4 = 0

       This equation involves a variable raised to the power of 2 and is more complex than a simple linear equation.

   3. Linear Equation in Two Variables:-

         2x + 3y = 6

       This equation involves two variables, x and y, and represents a straight line when graphed.    


Types Of Equation:-

   1. Linear Equations:-

      Equations of the first degree, where the highest power of the variable is 1.

      Example:-  3x + 2 = 11

   2. Quadratic Equations:-

       Equations where the highest power of the variable is 2.

       Example:-   x²− 5x + 6 = 0

   3. Polynomial Equations:-

       Equations involving terms with variables raised to positive integer powers.

       Example:-  2x³− x²+ 4x − 8 = 0

   4. Rational Equations:-

       Equations that involve ratios of polynomials.

                       (2x + 1)

       Example:-  ----------- = 4

                       (x − 3)

   5. Exponential Equations:-

       Equations where the variable appears in the exponent.

       Example:-  x² = 8

   6. Logarithmic Equations:-

       Equations that involve logarithms.

       Example:-  log⁡(x) + log⁡(2) = log⁡(8)

       The goal in solving an equation is to find the value(s) of the variable(s) that make the equation true. This is often done through algebraic manipulation, such as:


Solving The Equations:-

  • Isolating the variable:-  Rearranging the equation to get the variable on one side of the equation and the constants on the other.

         x + 3 = 7 (Subtract 3 from both sides)

         x + 3 - 3 = 7 - 3

  Or,  x = 4

  • Factoring:-  Breaking down a complex equation into simpler factors.
  • Using the Quadratic Formula:-   For solving quadratic equations 

                           - b ± √(b² - 4ac)

                    x = -------------------

                                  2a

  • Graphical Methods:-

       Plotting the equation and identifying points of intersection or other relevant features.


Real-World Applications:-

  • Physics:- Equations describe the relationships between physical quantities, like force and motion.
  • Economics:- Equations model relationships between economic variables like supply and demand.
  • Engineering:- Equations are used to design and analyze systems, like circuits or structures.

In essence, equations are tools for describing and analyzing relationships, patterns, and phenomena in both mathematics and the real world. They allow us to express and solve problems systematically.