Q2(ii):

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

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.

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 $3.14$).

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