Introduction to Linear Equations

What is an Equation?

An equation is a statement that two expressions are equal. It always has an = sign.

Example: \( 5x = 25 \) is an equation. - Left-hand side (LHS): \( 5x \) - Right-hand side (RHS): \( 25 \) The equation says both sides are the same when \( x \) has a certain value.

What is a Linear Equation?

A linear equation is an equation where the variable has the highest power of 1.
Examples of linear equations in one variable: \( 2x + 3 = 7 \), \( y - 5 = 10 \)

- It does not contain \( x^2, y^2, x^3 \), etc. - In Class 8, we mostly study equations in one variable.

Parts of an Equation

Every equation has two parts: - Left Hand Side (LHS) - Right Hand Side (RHS) Example: In \( 3x + 2 = 7 \), LHS: \( 3x + 2 \) RHS: \( 7 \)

Equation vs Expression

- An expression has numbers and variables but no equal sign. Example: \( 2x + 5, \, 7y - 3 \). - An equation has an equal sign and shows equality. Example: \( 2x + 5 = 11 \).

What is the Solution of an Equation?

The solution of an equation is the value of the variable that makes both sides equal.

Example: Solve \( 2x - 3 = 7 \)

👉 Add 3 to both sides: \( 2x = 10 \)

👉 Divide both sides by 2: \( x = 5 \)

So, the solution is \( x = 5 \).

How Do We Solve Equations?

- We can use addition, subtraction, multiplication, division on both sides to keep the equation balanced. - Moving a term from one side to another changes its sign. (This process is called transposing.)

Example Problem

Solve: \( x - 7 = 11 \)

👉 Add 7 to both sides: \( x = 18 \)

So, the solution is \( x = 18 \).

Summary: - An equation shows equality between two expressions. - A linear equation has variables with power 1 only. - Solving an equation means finding the value of the variable that balances both sides.