NCERT Class 9 Ganita Manjari Part 2 Chapter 13 Two Variables One Line PDF Download

Chapter 13: Two Variables, One Line

In Chapter 2 of Ganita Manjari, Part I, we explored linear patterns and learnt about linear polynomials. In this chapter, we will go a step further in exploring linear equations in two variables, by learning how to solve a pair of linear equations.

13.1 Linear Equations in Two Variables

Radha stops at a fruit cart. The price tags say that mangoes cost Rs. 60 per kg and bananas cost Rs. 50 per kg. She buys some kilograms of mangoes and some kilograms of bananas. Altogether, she pays Rs. 280. Can we express this situation as an equation?

Since we do not know how many kilograms of mangoes or bananas she bought, let us assume that

\( x \) = number of kilograms of mangoes

\( y \) = number of kilograms of bananas

The cost of the mangoes is Rs. 60\( x \) and the cost of the bananas is Rs. 50\( y \). The total cost is Rs. 280. Hence, we can express this using the linear equation

\[ 60x + 50y = 280 \]

Notice that the left-hand side (LHS) and the right-hand side (RHS) of the equation both represent the same quantity, that is, the total cost (in Rs.).

The equation \( 60x + 50y = 280 \) is a linear equation in two variables.

[Figure: A fruit vendor's cart with an umbrella, See in your textbook]

Teacher's Note

When setting up an equation from a word problem, first identify what you do not know and assign a variable to each unknown. Here, you do not know the kilograms of mangoes or bananas, so \( x \) and \( y \) are the right choice. Always check that your equation makes sense: does \( 60x + 50y = 280 \) correctly represent the total cost? Yes, because it adds the cost of mangoes plus the cost of bananas.

13.1.1 Standard Form of Linear Equations in Two Variables

Any equation that can be expressed 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.

\( ax + by + c = 0 \) is referred to as the standard form of the linear equation. Here \( a \) and \( b \) are the coefficients of the variables \( x \) and \( y \) respectively and \( c \) is a constant.

It is useful to note the following:

(i) The coefficients \( a \) and \( b \) and the constant \( c \) can also be rational numbers. For example, \( \frac{1}{2}x + \frac{3}{5}y - \frac{1}{2} = 0 \) is also a linear equation in two variables in standard form. However, for convenience, in such cases we often multiply the equation by a suitable number to remove the fractions and write the equation with integer coefficients. Here we can multiply by 20 to get the equation \( 5x + 12y - 10 = 0 \).

(ii) Equations of the type \( ax + c = 0 \) are special cases of linear equations in two variables as they can be expressed as \( ax + 0y + c = 0 \). For example, \( 7 + 4x = 0 \) can be written as \( 4x + 0y + 7 = 0 \), with \( a = 4 \), \( b = 0 \) and \( c = 7 \).

Example 1: Write each of the following equations in standard form, \( ax + by + c = 0 \), and indicate the values of \( a \), \( b \) and \( c \) in each case:

(i) \( 5x + 2y = 2.7 \)     (ii) \( \frac{x}{2} - 4 = \frac{3}{2}y \)     (iii) \( 4 = 1.5x - 1.3y \)     (iv) \( 3y = 5 \)

Solution:

(i) \( 5x + 2y = 2.7 \) can be written as \( 5x + 2y - 2.7 = 0 \). Here \( a = 5 \), \( b = 2 \) and \( c = - 2.7 \). The equation can also be rewritten as \( -5x - 2y + 2.7 = 0 \). In this case \( a = -5 \), \( b = -2 \), \( c = 2.7 \).

(ii) \( \frac{x}{2} - 4 = \frac{3}{2}y \) can be written as \( \frac{x}{2} - \frac{3}{2}y - 4 = 0 \). Here \( a = \frac{1}{2} \), \( b = - \frac{3}{2} \) and \( c = -4 \).

(iii) \( 4 = 1.5x - 1.3y \) can be written as \( 1.5x - 1.3y - 4 = 0 \). Here \( a = 1.5 \), \( b = -1.3 \) and \( c = -4 \). It can also be written as \( -1.5x + 1.3y + 4 = 0 \). In this case \( a = -1.5 \), \( b = 1.3 \) and \( c = 4 \).

(iv) \( 3y = 5 \) can be written as \( 0x + 3y - 5 = 0 \). Here \( a = 0 \), \( b = 3 \) and \( c = -5 \).

Teacher's Note

Notice that the standard form is not unique: you can multiply the entire equation by \( -1 \) and get different values for \( a \), \( b \), and \( c \). Both \( 5x + 2y - 2.7 = 0 \) and \( -5x - 2y + 2.7 = 0 \) are correct. In exams, either form is acceptable unless the question asks for a specific constraint, like positive coefficients.

Exercise Set 13.1

1. Write a linear equation in two variables in which \( a = 3 \), \( b = 0 \) and \( c = -\frac{1}{5} \).
2. Complete the following table after expressing the given linear equations in standard form.

   Linear equation in two variables	Standard Form	Coefficient of \( x \)	Coefficient of \( y \)	Constant term
   \( y - 15 = \sqrt{2}x \)
   \( 3y - 2x = 0 \)
   \( 5x = 3y \)
   \( x = 8 \)
   \( 3y = 1 \)

3. (i) The cost of a notebook is twice the cost of a pen. Consider the cost of a notebook to be Rs. \( t \) and that of a pen to be Rs. \( p \). Charlie wrote \( t = 2p \), whereas Meera wrote \( p = 2t \). Which of these two representations is correct?

   (ii) In a one-day International Cricket match between India and Sri Lanka played in Nagpur, two Indian batsmen together scored 176 runs. Manisha expressed this situation as \( x + y = 176 \), where the number of runs scored by one batsman is \( x \), and the number of runs scored by the other is \( y \). Is this a correct representation?

Key Points

• A linear equation in two variables has the form \( ax + by + c = 0 \), where \( a \), \( b \), and \( c \) are real numbers and at least one of \( a \) or \( b \) is not zero.
• The standard form is not unique: you can multiply the entire equation by any non-zero number and still have the same equation, even though \( a \), \( b \), and \( c \) change.
• Equations with only one variable, like \( x = 8 \) or \( 3y = 5 \), are special cases of linear equations in two variables where one coefficient is zero.
• When converting a word problem to an equation, define your variables clearly for the quantities you do not know, then write the relationship between them.

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