NCERT Solutions Class 9 Mathematics Ganita Manjari 2 Chapter 13 Two Variables One Line Exercise 13.1

Exercise Set 13.1

Question 1. Write a linear equation in two variables in which a = 3, b = 0 and c = \( -\frac{1}{5} \).
Answer:
Put the values into ax + by + c = 0:
3x + 0 × y + (\( -\frac{1}{5} \)) = 0, that is, 3x + 0 × y − \( \frac{1}{5} \) = 0.
Clearing the fraction (multiply by 5): 15x + 0 × y − 1 = 0.
Required equation: 3x + 0·y − \( \frac{1}{5} \) = 0 (equivalently 15x + 0·y − 1 = 0).

Question 2. Complete the following table after expressing the given linear equations in standard form. (Columns: Linear equation in two variables | Standard form | Coefficient of x | Coefficient of y | Constant term.) The equations are: y − 15 = √2x; 3y − 2x = 0; 5x = 3y; x = 8; 3y = 1.
Answer:
Move every term to the left, writing the x-term first, then the y-term, then the constant.
1. y − 15 = √2x → y − 15 − √2x = 0 → (× −1) √2x − y + 15 = 0. Coefficient of x = √2, of y = −1, constant = 15.
2. 3y − 2x = 0 → −2x + 3y + 0 = 0. Coefficient of x = −2, of y = 3, constant = 0.
3. 5x = 3y → 5x − 3y + 0 = 0. Coefficient of x = 5, of y = −3, constant = 0.
4. x = 8 → x + 0·y − 8 = 0. Coefficient of x = 1, of y = 0, constant = −8.
5. 3y = 1 → 0·x + 3y − 1 = 0. Coefficient of x = 0, of y = 3, constant = −1.
Note: multiplying any row by −1 also gives a correct standard form, e.g. row 1 as −√2x + y − 15 = 0 (coefficients −√2, 1, −15) and row 2 as 2x − 3y + 0 = 0 (coefficients 2, −3, 0).

Teacher's Note:
1. A missing variable still gets a place in the table, with coefficient 0.
2. Signs change when a term crosses the equals sign; this is where most mistakes happen.

Question 3. (i) The cost of a notebook is twice the cost of a pen. Consider the cost of a notebook to be ₹t and that of a pen to be ₹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?
Answer:
(i) 'Notebook costs twice a pen' means cost of notebook = 2 × cost of pen, i.e. t = 2p.
Test with numbers: pen ₹5, notebook ₹10. Then t = 2p gives 10 = 10 ✓, but p = 2t gives 5 = 20 ✗.
Charlie's equation t = 2p is correct. Meera's p = 2t would mean the pen costs twice the notebook.
(ii) Runs of one batsman = x, of the other = y. Together they made 176, so x + y = 176.
Yes, Manisha's representation is correct. In standard form it is x + y − 176 = 0, with a = 1, b = 1, c = −176.