Q2(i):

Write four solutions for each of the following equations: (i) $2x + y = 7$

Initial Setup & Algebraic Isolation

We are given the linear equation in two variables:

$2x + y = 7$

To systematically find solutions for this equation, it is mathematically highly efficient to express one variable in terms of the other. We will isolate the dependent variable $y$ on the left side of the equation [Per the fundamental properties of equality, by subtracting $2x$ from both sides].

$y = 7 - 2x$

A "solution" to a linear equation in two variables is an ordered pair $(x, y)$ that satisfies the equation. Since this is a linear equation defined over the set of real numbers ($\mathbb{R}$), it possesses infinitely many solutions. We will select four arbitrary integer values for the independent variable $x$ to compute the corresponding values of $y$.

Step 1: Calculating the First Solution

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

$y = 7 - 2(0)$

$y = 7 - 0$

$y = 7$

This yields our first ordered pair: $(0, 7)$.

Step 2: Calculating the Second Solution

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

$y = 7 - 2(1)$

$y = 7 - 2$

$y = 5$

This yields our second ordered pair: $(1, 5)$.

Step 3: Calculating the Third Solution

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

$y = 7 - 2(2)$

$y = 7 - 4$

$y = 3$

This yields our third ordered pair: $(2, 3)$.

Step 4: Calculating the Fourth Solution

Let us assign the value $x = 3$. Substituting this into the equation:

$y = 7 - 2(3)$

$y = 7 - 6$

$y = 1$

This yields our fourth ordered pair: $(3, 1)$.

Tabular Representation of Solutions

We can organize the derived coordinates into a data table to clearly demonstrate the linear relationship between $x$ and $y$. Notice that for every $1$ unit increase in $x$, $y$ decreases by $2$ units [This represents the slope of the line, $m = -2$].

Graphical Representation (Cartesian Plane)

Plotting these four points on a Cartesian coordinate system verifies that they are collinear, forming the straight line defined by $2x + y = 7$.

Final Solution: Four distinct solutions for the linear equation $2x + y = 7$ are $(0, 7)$, $(1, 5)$, $(2, 3)$, and $(3, 1)$.