Q2(iii):

Write four solutions for each of the following equations: (iii) $x = 4y$

Initial Setup & Theoretical Foundation

We are given the linear equation in two variables:

$x = 4y$

[Per the fundamental theorem of linear algebra], a linear equation in two variables of the form $ax + by + c = 0$ has infinitely many solutions. This is because for every arbitrary real value assigned to the independent variable, there exists a unique corresponding real value for the dependent variable. To find four distinct solutions, we will systematically assign four different real values to $y$ and solve for the corresponding values of $x$.

Step 1: Deriving the First Solution

Let us assign the value $y = 0$. Substituting this into the given equation:

$x = 4(0)$

$x = 0$

Thus, the first ordered pair $(x, y)$ that satisfies the equation is $(0, 0)$.

Step 2: Deriving the Second Solution

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

$x = 4(1)$

$x = 4$

Thus, the second ordered pair $(x, y)$ that satisfies the equation is $(4, 1)$.

Step 3: Deriving the Third Solution

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

$x = 4(2)$

$x = 8$

Thus, the third ordered pair $(x, y)$ that satisfies the equation is $(8, 2)$.

Step 4: Deriving the Fourth Solution

Let us assign a negative integer to demonstrate solutions in the third quadrant. Let $y = -1$. Substituting this into the given equation:

$x = 4(-1)$

$x = -4$

Thus, the fourth ordered pair $(x, y)$ that satisfies the equation is $(-4, -1)$.

Tabular Representation of Solutions

We can organize the derived solutions into a coordinate table to clearly map the relationship between $x$ and $y$.

Graphical Verification

Plotting these points on a Cartesian plane confirms that they are collinear, forming the straight line defined by $x = 4y$.

Final Solution: Four valid solutions for the equation $x = 4y$ are $(0, 0)$, $(4, 1)$, $(8, 2)$, and $(-4, -1)$.