Q1:

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

Initial Setup & Theoretical Foundation

The given mathematical statement is the equation $y = 3x + 5$. This is a linear equation in two variables, $x$ and $y$. The standard form of a linear equation in two variables is $ax + by + c = 0$. By rearranging the given equation, we obtain $-3x + y - 5 = 0$, where the coefficients are $a = -3$, $b = 1$, and the constant term is $c = -5$.

The standard options for this classical problem are:

• (i) A unique solution
• (ii) Only two solutions
• (iii) Infinitely many solutions

Step 1: Algebraic Analysis of Solutions

A "solution" to a linear equation in two variables is defined as an ordered pair $(x, y)$ that satisfies the equation. In the equation $y = 3x + 5$, $x$ acts as the independent variable and $y$ acts as the dependent variable.

[Per the fundamental properties of real numbers], for every distinct real value assigned to $x$, there exists a corresponding unique real value for $y$. Let us systematically substitute arbitrary real numbers for $x$ to observe the resulting values of $y$:

Because the set of real numbers ($\mathbb{R}$) is infinite, we can choose infinitely many values for $x$. Consequently, we will generate infinitely many corresponding values for $y$. Therefore, the equation possesses infinitely many solutions.

Step 2: Geometric Interpretation

[By the geometric definition of linear equations], any linear equation in two variables represents a straight line on a two-dimensional Cartesian coordinate system. Every single point that lies on this line represents a valid solution $(x, y)$ to the equation.

A straight line extends indefinitely in both directions. Since a line is composed of an infinite number of points, the equation representing that line must inherently have an infinite number of solutions.

Step 3: Visual Proof via Cartesian Mapping

Below is the precise geometric representation of the equation $y = 3x + 5$. The red coordinates highlight specific solutions, demonstrating that the line passes through an infinite continuum of points.

Logical Deduction

Based on both the algebraic substitution method and the geometric mapping, it is evident that the relationship $y = 3x + 5$ is not restricted to one or two specific coordinate pairs. The equation acts as a continuous function mapping the domain of all real numbers to a range of all real numbers.

Final Solution: The true option is "Infinitely many solutions". This is because for every real value of $x$, there is a corresponding unique real value of $y$, and geometrically, the equation represents a straight line composed of an infinite number of points.