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Q2(ii):
Write four solutions for each of the following equations: (ii) $\pi x + y = 9$

Solution :

Initial Setup & Algebraic Formulation

We are tasked with finding four distinct solutions for the following linear equation in two variables:

$\pi x + y = 9$

A linear equation in two variables has infinitely many solutions. A solution is defined as an ordered pair $(x, y)$ that satisfies the equation. To systematically generate these solutions, we will express one variable in terms of the other.

Step 1: Isolating the Dependent Variable

To make the calculation of ordered pairs straightforward, we isolate the dependent variable ($y$) in terms of the independent variable ($x$). [Per the Subtraction Property of Equality, subtracting $\pi x$ from both sides of the equation yields:]

$y = 9 - \pi x$

We can now substitute arbitrary real numbers for $x$ to compute the corresponding exact values of $y$.

Step 2: Deriving the First Solution

Let us assign $x = 0$. Substituting this value into our isolated equation:

$y = 9 - \pi(0)$

$y = 9 - 0$

$y = 9$

Thus, the first solution is the ordered pair $(0, 9)$. [This point also represents the y-intercept of the linear graph].

Step 3: Deriving the Second Solution

Let us assign $x = 1$. Substituting this value into the equation:

$y = 9 - \pi(1)$

$y = 9 - \pi$

Thus, the second solution is the ordered pair $(1, 9 - \pi)$.

Step 4: Deriving the Third Solution

Let us assign $x = 2$. Substituting this value into the equation:

$y = 9 - \pi(2)$

$y = 9 - 2\pi$

Thus, the third solution is the ordered pair $(2, 9 - 2\pi)$.

Step 5: Deriving the Fourth Solution

Let us assign $x = -1$. Substituting this value into the equation:

$y = 9 - \pi(-1)$

$y = 9 + \pi$

Thus, the fourth solution is the ordered pair $(-1, 9 + \pi)$.

Step 6: Tabular Summary of Solutions

We can organize the derived solutions into a Cartesian coordinate table. This demonstrates the linear relationship and prepares the data for graphical representation.

Independent Variable ($x$) Dependent Variable ($y$) Ordered Pair $(x, y)$
$0$ $9$ $(0, 9)$
$1$ $9 - \pi$ $(1, 9 - \pi)$
$2$ $9 - 2\pi$ $(2, 9 - 2\pi)$
$-1$ $9 + \pi$ $(-1, 9 + \pi)$

Step 7: Graphical Representation of the Linear Equation

Below is the geometric interpretation of the equation $\pi x + y = 9$ plotted on a Cartesian plane. The line represents the locus of all infinite solutions, with our four specific calculated points highlighted. (Note: For spatial plotting purposes, $\pi$ is approximated as $3.14$).

x y 0 1 2 -1 9 (0, 9) (1, 9-π) (2, 9-2π) (-1, 9+π) πx + y = 9

Final Solution: Four valid solutions for the equation $\pi x + y = 9$ are $(0, 9)$, $(1, 9 - \pi)$, $(2, 9 - 2\pi)$, and $(-1, 9 + \pi)$.


More Questions from Class 9 Mathematics Linear Equations in Two Variables EXERCISE 4.2


CBSE Solutions for Class 9 Mathematics Linear Equations in Two Variables


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