NCERT Solutions Class 9 Maths Chapter 13 Two Variables, One Line on Padhaiwise

NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 13 Two Variables, One Line Exercise Set 13.1

Mon, 01 Jan 0001 00:00:00 +0000

NCERT Solutions for Class 9 Maths Chapter 13 — Exercise Set 13.1#

A linear equation in two variables in standard form is
$$ax + by + c = 0$$
where $a$, $b$, $c$ are real numbers and $a$, $b$ are not both zero. $a$ is the coefficient of $x$, $b$ the coefficient of $y$, and $c$ the constant term.

1. Write a linear equation in two variables in which a = 3, b = 0 and c = −1/5. #

Solution

Substituting in $ax + by + c = 0$:

NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 13 Two Variables, One Line Exercise Set 13.2

Mon, 01 Jan 0001 00:00:00 +0000

NCERT Solutions for Class 9 Maths Chapter 13 — Exercise Set 13.2#

A solution of a linear equation in two variables is an ordered pair $(x, y)$ that makes the equation true. Every linear equation in two variables has infinitely many solutions, and they are exactly the points of its graph — a straight line.

To find a solution: choose any value of $x$, substitute it, and solve for $y$.

NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 13 Two Variables, One Line Exercise Set 13.3

Mon, 01 Jan 0001 00:00:00 +0000

NCERT Solutions for Class 9 Maths Chapter 13 — Exercise Set 13.3#

$$\text{Slope} = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}$$
A larger slope means a steeper line. A horizontal line has slope 0; a vertical line has no defined slope.

1. The following diagrams represent ski hills. Rank the hills in order of their steepness, from least to greatest. (A: rise 60 m, run 70 m; B: rise 60 m, run 110 m; C: rise 80 m, run 100 m) #

Solution

$$\text{Slope of A} = \dfrac{60}{70} \approx 0.86$$

NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 13 Two Variables, One Line Exercise Set 13.4

Mon, 01 Jan 0001 00:00:00 +0000

NCERT Solutions for Class 9 Maths Chapter 13 — Exercise Set 13.4#

A pair of linear equations in two variables is written as

$$a_1x + b_1y + c_1 = 0, \qquad a_2x + b_2y + c_2 = 0$$

Each fact in the situation gives one equation.

Given below are some everyday situations. Construct a pair of linear equations in two variables for each situation.

1. On two different days a family buys movie tickets and snack boxes. Let the cost of one movie ticket be ₹x and the cost of one snack box be ₹y. (i) On the first day, the family buys 2 movie tickets and 3 snack boxes for ₹850. (ii) On the second day, the family buys 4 movie tickets and 1 snack box for ₹1100. #

Solution

On the first day, 2 tickets and 3 snack boxes cost ₹850, so:

NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 13 Two Variables, One Line Exercise Set 13.5

Mon, 01 Jan 0001 00:00:00 +0000

NCERT Solutions for Class 9 Maths Chapter 13 — Exercise Set 13.5#

For $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$:

Ratios Lines Solutions

$\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$ intersect exactly one

$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$ parallel none

$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$ coincident infinitely many

Ratios	Lines	Solutions
$\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$	intersect	exactly one
$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$	parallel	none
$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$	coincident	infinitely many

1. Form a pair of linear equations for each of the following problems and find their solutions. #

(i) The sum of two integers is +5 and their difference is −21. Find the two numbers.

Let the numbers be $x$ and $y$.

NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 13 Two Variables, One Line End-of-Chapter Exercises

Mon, 01 Jan 0001 00:00:00 +0000

NCERT Solutions for Class 9 Maths Chapter 13 — End-of-Chapter Exercises#

Slope-intercept form: $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept.

For a pair of equations: $\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$ → one solution; $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$ → none; $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$ → infinitely many.

1. The graph of the line y = 3x, passing through (0, 0) and (2, 6) is given below. #

(i) Identify the slope of the line from the graph and explain how you calculated it using the two points on the line.

From $(0, 0)$ to $(2, 6)$, the line rises 6 while running 2: