# Linear Equations In Two Variables - Class 9 - Mathematics

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## Exercise 4.1 - Linear Equations In Two Variables | NCERT | Mathematics | Class 9

The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement.

(Take the cost of a notebook to be ₹ $x$ and that of a pen to be ₹ $y$ ).

Given the cost of a notebook is represented by ₹ $x$ and the cost of a pen by ₹ $y$, the relationship described can be summarized as:

"The cost of a notebook is twice the cost of a pen."

Translating this into a linear equation in two variables, we get:

$$ x = 2y $$

Here, $x$ and $y$ are the variables representing the cost of a notebook and a pen, respectively.

Express the following linear equations in the form $a x+b y+c=0$ and indicate the values of $a, b$ and $c$ in each case:

(i) $2 x+3 y=9.3 \overline{5}$

(ii) $x-\frac{y}{5}-10=0$

(iii) $-2 x+3 y=6$

(iv) $x=3 y$

(v) $2 x=-5 y$

(vi) $3 x+2=0$

(vii) $y-2=0$

(viii) $5=2 x$

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Sign up now## Exercise 4.2 - Linear Equations In Two Variables | NCERT | Mathematics | Class 9

Which one of the following options is true, and why? $y=3 x+5$ has

(i) a unique solution,

(ii) only two solutions,

(iii) infinitely many solutions

The equation $y = 3x + 5$ represents a straight line in the XY-plane. For any unique value of $x$, there is a unique corresponding value of $y$. This relationship between $x$ and $y$ implies that for every point $(x, y)$ that satisfies the equation, $x$ determines $y$ uniquely. Therefore, for each input value of $x$, there is exactly one output value for $y$.

Given that a straight line extends infinitely in both directions along the XY-plane, there are infinitely many points $(x, y)$ that satisfy the equation $y = 3x + 5$. Each of these points represent a unique combination of $x$ and $y$ that fits the equation, and since the line is continuous and unbounded, it crosses every possible $x$-value exactly once, yielding a corresponding $y$-value.

**Therefore**, the correct option is:

(iii) infinitely many solutions

The reason is that the equation depicts a linear relationship with a constant slope, indicating that for every value of $x$, there is a unique value of $y$, and since $x$ can assume any value from the set of real numbers, there are infinitely many such $(x, y)$ pairs that satisfy the equation.

Write four solutions for each of the following equations:

(i) $2 x+y=7$

(ii) $\pi x+y=9$

(iii) $x=4 y$

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Sign up nowCheck which of the following are solutions of the equation $x-2 y=4$ and which are not:

(i) $(0,2)$

(ii) $(2,0)$

(iii) $(4,0)$

(iv) $(\sqrt{2}, 4 \sqrt{2})$

(v) $(1,1)$

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Sign up nowFind the value of $k$, if $x=2, y=1$ is a solution of the equation $2 x+3 y=k$.

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## Extra Questions - Linear Equations In Two Variables | NCERT | Mathematics | Class 9

The radius of a circle is proportional to its diameter. Find the constant of proportionality.

A) $\frac{1}{2}$

B) 5

C) $2 \pi$

D) $\pi$

The relationship between the radius and the diameter of a circle is governed by the equation:

$$ \text{Radius} = \frac{1}{2} \times \text{Diameter} $$

Here, the **constant of proportionality** between the radius and the diameter is **$\frac{1}{2}$**. This constant represents the fraction of the diameter that equals the radius.

Thus, the correct answer is:

A) $\frac{1}{2}$

Give two solutions for $3y+4=0$.

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The slope of the line passing through points $(3,2)$ and $(2,5)$ is:

A) $\frac{1}{3}$ B) -3 C) $-\frac{1}{3}$ D) 3