This document provides an introduction to linear equations in two variables. It defines a linear equation in two variables as one that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not both zero. Examples are given of writing equations in this form and identifying the values of a, b, and c. The document also discusses that a linear equation in two variables has infinitely many solutions, which can be represented as ordered pairs (x,y) that satisfy the equation.

1. CLASS-9
SUBJECT-MATHEMATICS
CHAPTER-04 (Lecture-01)
LINEAR EQUATIONS IN TWO
VARIABLES

2. Learning Objectives
*Introduction
*Solution of a Linear Equation
*Graph of Linear Equation in Two Variable
*Equation of Lines Parallel to the x- axis y-axis

3. INTRODUCTION
Lets recall what we have studied so far.
2x+5=0 ⇒ 2𝑥 = −5 ⇒ 𝑥 = −
5
2
Its Solution is -5/2 or we can say root of equation is -5/2.

4. Let us now consider the following situation:
In a One-day International Cricket match between India and Sri Lanka played in
Nagpur, two Indian batsmen together scored 176 runs. Express this information
in the form of an equation.
x+ y=176 which is the required equation.
This is an example of a linear equation in two variables.
So, any equation which can be put in the form ax + by + c = 0, where a, b and c
are real numbers, and a and b are not both zero, is called a linear equation in
two variables.

5. Example 1 : Write each of the following equations in the form ax + by + c = 0 and
indicate the values of a, b and c in each case:
(i) 2x + 3y = 4.37
Sol: 2x + 3y + (-4.37)= 0
Now comparing with ax + by + c = 0
We have a=2, b=3 and c= -4.37
(ii) x – 4 = 3 y
Sol: x+(-3)y+(-4)= 0 a=1, b=-3 and c= -4

6. Example 2 : Write each of the following as an equation in two variables:
(i) x = –5
Sol: x+ 0.y= -5
(ii) y = 2
Sol: 0.x+y=2

7. EXERCISE 4.1
1. The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent
this statement.
Sol: Let the cost of notebook to be Rs.x and that of a pen be Rs.y
x=2y
Solve by Yourself
2. Express the following linear equations in the form ax + by + c = 0 and indicate the values of a, b and c
in each case:
(i) 2x + 3y = 9.35
Sol: 2x+3y-9.35=0
Comparing to ax + by + c = 0 we have
a=2,b=3 and c=9.35

8. (iii) 2x = –5y
Sol: 2x+5y=0
a=2, b=5 and c=0
(iv) y – 2 = 0
Sol: 0.x+y-2=0
a=0,b=1 and c=-2

9. Qu. The cost of a pen is Rs.5 less than half of the cost of a fountain pen. Write
this statement as a linear equation in two variables.
Sol: Here, here variable quantities are cost of pen and cost of fountain pen.
So, let cost of a pen = Rs.x
and cost of fountain pen =Rs.y
According the question,
Cost of pen = half cost of fountain pen−5
𝑥 =
𝑦
2
− 5
⇒ 𝑥 =
𝑦 − 10
2
⇒ 2𝑥 = 𝑦 − 10
⇒ 2𝑥 − 𝑦 + 10 = 0
Which is required linear equation in two variables.

10. Qu. A lending library has a fixed charge for the first three days and an additional
charge for each day thereafter. Roshni paid Rs.35 for a book kept for seven days.
Write this statement as a linear equation in two variables.
Sol: Let the fixed charge be Rs.x and thereafter the additional charge be Rs.y per
day
Then, according to the question,
𝑥 + 7 − 3 𝑦 = 35
𝑥 + 4𝑦 = 35
Which is required linear equation in two variables.

11. We have seen that every linear equation in one variable has a unique solution.
What can we say about the solution of a linear equation involving two variables?
Let us consider the equation 2x + 3y = 12.
This solution is written as an ordered pair (3, 2), first writing the value for x and
then the value for y. Similarly, (0, 4) is also a solution for the equation above.
On the other hand, (1, 4) is not a solution of 2x + 3y = 12,because on
putting x = 1 and y = 4 we get 2x + 3y = 14, which is not 12.
Note that (0, 4) is a solution but not (4, 0).

12. You have seen at least two solutions for 2x + 3y = 12, i.e., (3, 2) and (0, 4).
Can you find any other solution?
Do you agree that (6, 0) is another solution? Verify the same. In fact, we can get many
many solutions in the following way.
2 × 6 + 3 × 0 = 12 ⇒ 12 + 0 = 12 ⇒ 12 = 12
Hence (6,0) is a solution.
A linear equation in two variables has infinitely many solutions

13. Thank You