Exercise 4.4: Linear Equations in Two Variables — NCERT Solutions

A line parallel to the $x$-axis has every point with the same $y$-coordinate. Its equation is of the form $y = k$.

Since the line passes through $(2, -3)$, the constant $y$-value is $-3$:

$y = -3$

Graph: Draw a horizontal straight line crossing the $y$-axis at $(0, -3)$ and running parallel to the $x$-axis. It passes through points such as $(0,-3)$, $(2,-3)$ and $(-4,-3)$.

The equation $x = 4$ can be written in two variables as $x + 0\cdot y = 4$. Here $x$ is always $4$, while $y$ can be any real number.

Some solutions:

| $x$ | $y$ | Point | |-----|-----|-------| | $4$ | $0$ | $(4, 0)$ | | $4$ | $2$ | $(4, 2)$ | | $4$ | $-1$ | $(4, -1)$ |

Plot these points and join them — they form a vertical straight line parallel to the $y$-axis, cutting the $x$-axis at $(4, 0)$.

Such a line ($x = k$) is always parallel to the $y$-axis.

Solve for $y$:

$2y + 5 = 0 \Rightarrow 2y = -5 \Rightarrow y = -\tfrac{5}{2}$

(i) In one variable: On the number line, $y = -\tfrac{5}{2}$ is a single point located at $-2.5$.

(ii) In two variables: Written as $0\cdot x + 2y + 5 = 0$, the value $y = -\tfrac{5}{2}$ holds for every value of $x$. This is a horizontal straight line parallel to the $x$-axis, passing through $(0, -\tfrac{5}{2})$.

So the same equation represents a point on a line, but a whole line in the plane.