Chapter 4 - Linear Equations in Two Variables, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 PDF Download

EQUATION
A statement of equality which contain one or more unknown quantity or variable (literals) is called an equation.
Ex. : 2x - 5 = 23,

An equation has two parts. The part which is on the left side to the equality sign is known as left hand side
(L.H.S) and the part which is on the right side to the equality sign is known as right hand side (R.H.S).

Consider an equation

Variable : The unknown quantities used in any equation are knows as variables. Generally, they are denoted
by the last English alphabets x, y, z etc.

Linear Equation : An equation in which the maximum power of variable is one is called a linear equation.
Ex. :- 4x + 5 = 3x + 1, 2x + 3y = 4 are linear equations.

Linear Equation in one variable : In general the equation of the form ax + b = c where, a, b and c
are real numbers and a  0 is called linear equations in one variable.

Solution of linear equations in one variable : The value of the variable which when substituted in an equation, makes L.H.S. = R.H.S. is said to satisfy the equation is called a solution or a root of the equation.
  is a solution of the equation
The standard form of the linear equation in one variable is ax + b = 0, where a and b are real numbers and a  0.

Remarks :
(i) Linear equation in one variable has a unique (one and only one) solution.
(ii) We can add or subtract same number from each side of an equation.
(iii) We can multiply or divide both the sides of an equation by same non-zero number.

Ex 1. Verify that x = 4 is a solution of the equation 2x - 3 = 5.
 Sol. Substituting x = 4 in the given equation, we get
L.H.S. = 2x - 3 = 2 × 4 - 3 = 8 - 3 = 5 = R.H.S.
Hence, x = 4 is a solution of the equation 2x - 3 = 5

Ex 2. Solve: 3x + 2 = 11
 Sol. 3x + 2 = 11

Hence, x = 3 is the solution of the given equation.

RULES FOR SOLVING A LINEAR EQUATION IN ONE VARIABLE

Rule-I : Same quantity (number) can be added to both sides of an equation without changing the equality 

Ex 3. Solve : x - 3 = 4
 Sol. ⇒ x - 3 + 3 = 4 + 3 (equal number is added on both sides)
x = 7

Rule-II : Same quantity (number) can be subtracted from both sides of an equation without changing
 the equality.

Ex 5. :Solve : x + 5 = 9
 Sol. ⇒ x + 5 - 5 = 9 - 5 (equal number 5 is subtracted from both sides)
⇒ x = 4
Thus, x = 4 is the solution of the given equation.

Rule-III : Both sides of an equation may be multiplied by the same non-zero number without changing
 the equality.

Rule-IV : Both sides of an equation may be divided by the same non-zero number without changing
 the equality.

Ex 8. Solve : 2x = 7

Rule-V : (Transposition) If any term of an equation is taken from one side to the other side then the
 sign changes. This process is called transposition.

Ex 9. 10x - 27 = 7 - 7x

Sol.

Ex 10. Solve 3x - 7 = 17

Sol. 

Rule-VI : (Cross multiplication method) : If acxx ++ db = mn then n(ax + b) = m (cx + d). This is called
 cross-multiplication method.