Graphical Method of Solution Questions in English

(D) line parallel to the $X$-axis is of the form $y = k$,where $k$ is a constant.
This is because for any value of $x$,the $y$-coordinate remains the same,resulting in a horizontal line.
Analyzing the given options:
$A$) $x=y$ represents a line passing through the origin with a slope of $1$.
$B$) $2x-y=0$ or $y=2x$ represents a line passing through the origin with a slope of $2$.
$C$) $x-2y=0$ or $y=x/2$ represents a line passing through the origin with a slope of $0.5$.
$D$) $2y=3$ can be written as $y = 1.5$,which is a constant value for $y$ regardless of $x$.
Therefore,the graph of $2y=3$ is a horizontal line parallel to the $X$-axis.

(A) linear equation in two variables $ax + by + c = 0$ passes through the origin $(0, 0)$ if and only if the constant term $c$ is equal to $0$.
In the given options:
$A) 4x = 5y \implies 4x - 5y = 0$. Here,$c = 0$,so it passes through the origin.
$B) 4x + 1 = 5y \implies 4x - 5y + 1 = 0$. Here,$c = 1 \neq 0$.
$C) 5x - 1 = 4y \implies 5x - 4y - 1 = 0$. Here,$c = -1 \neq 0$.
$D) 4x + 5y = 1 \implies 4x + 5y - 1 = 0$. Here,$c = -1 \neq 0$.
Therefore,the equation $4x = 5y$ passes through the origin.

(C) From the given graph,it is observed that the two lines are parallel to each other.
Parallel lines never intersect each other at any point.
Since the solution of a pair of linear equations is the point of intersection of the lines,and these lines do not intersect,there is no common point between them.
Therefore,the pair of equations represented by these lines has no solution.

(A) In the given figure,the two lines intersect each other at a single point.
According to the geometric interpretation of linear equations in two variables,if two lines intersect at a single point,the pair of linear equations has a unique solution (exactly one solution).
Therefore,the correct option is $A$.

(B) When two lines are coincident,every point on the line is a common solution to both equations.
Since a line consists of an infinite number of points,the pair of linear equations has infinitely many solutions.
Therefore,the correct option is $B$.