Chapter 4 - Linear Equations in Two Variables, Solved Examples, Class 9, Maths

# 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.

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## FAQs on Chapter 4 - Linear Equations in Two Variables, Solved Examples, Class 9, Maths - Extra Documents & Tests for Class 9

 1. What is a linear equation in two variables?
Ans. A linear equation in two variables is an equation that can be expressed in the form ax + by = c, where a, b, and c are constants and x and y are variables. The graph of a linear equation in two variables is a straight line.
 2. How do you solve a linear equation in two variables?
Ans. To solve a linear equation in two variables, we can use either the substitution method or the elimination method. In the substitution method, we solve one equation for one variable and substitute it into the other equation. In the elimination method, we add or subtract the equations to eliminate one variable and solve for the other.
 3. Can a linear equation in two variables have more than one solution?
Ans. Yes, a linear equation in two variables can have infinitely many solutions or no solution. If the graph of the equation represents a straight line, then there are infinitely many solutions. If the graph represents parallel lines, then there is no solution. If the graph represents intersecting lines, then there is exactly one solution.
 4. What is the importance of linear equations in two variables?
Ans. Linear equations in two variables are important in many areas of mathematics and real-life applications. They help us represent and solve problems involving two related variables, such as distance and time, cost and quantity, or temperature and pressure. They provide a way to model and analyze various situations in fields like physics, economics, engineering, and more.
 5. Can linear equations in two variables be represented graphically?
Ans. Yes, linear equations in two variables can be represented graphically. The graph of a linear equation in two variables is a straight line. By plotting points on the coordinate plane and connecting them, we can obtain the graph of the equation. The slope-intercept form (y = mx + b) of the equation provides information about the slope and y-intercept, which helps in sketching the graph. Graphical representation helps in visualizing the relationship between the variables and understanding the solutions to the equation.

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