Question

Find the equation of a straight line parallel to the $x$- axis.
(A) $x=a$
(B) $y=a$
(C) $y=x$
(D) $y=-a$

Answer 1

Hint: Express the equation of$x$- axis in the form$Ax+By+C=0$ and find its slope using the formula $m={\displaystyle \frac{-A}{B}}$ where m represents the slope.
Use the fact that parallel lines have the same slope to calculate the slope of the line parallel to the $x$- axis. Use this slope and the general equation of the parallel line $Ax+By+C_1=0$ where $C\ne C_1$ to get the answer.

Complete step by step answer:
We are asked to find the equation of a straight line which is parallel to the $x$- axis.
We know that the equation of the $x$- axis is$y=0$
This means that any point on the $x$- axis will have its $y-$ coordinate as 0.
Any line which is parallel to the $x$- axis will look like the below figure:

That is, the line will lie either above the $x$- axis or below it. Also, it will pass through the $y$- axis.
The general equation of a straight line is given by $Ax+By+C=0$
Here, the letters A, B, and C are real numbers. Also, A and B are non-zero constants.
Now, the slope of the straight line given by the equation $Ax+By+C=0$is $m={\displaystyle \frac{-A}{B}}$ where $B\ne 0$
We will compare the equation of $x$- axis with the general form $Ax+By+C=0$to obtain its slope.
Now, the equation of $x$- axis is $y=0$.
Therefore, we can express this equation as $0x+y+0=0$.
Thus on comparison with the general form $Ax+By+C=0$, we get $A=0$,$B=1$ and $C=0$
Therefore, the slope of the equation of $x$- axis is $m=-{\displaystyle \frac{0}{1}}=0$
But the question is not about finding the slope or equation of the $x$- axis.
We need the equation of the line parallel to the $x$- axis.
Now, we know that the general equation of any line which is parallel to the line represented by the equation $Ax+By+C=0$ would be of the form $Ax+By+C_1=0$ where $C\ne C_1$ and $C_1$ is also a real number.
We can notice here that the coefficients A and B of the variables $x$ and $y$ are the same for a pair of parallel lines.
Therefore, the slope of a line parallel to the line represented by the equation $Ax+By+C=0$will also be given by $m={\displaystyle \frac{-A}{B}}$ where $B\ne 0$
Thus, we can conclude that parallel lines have the same slope.
Therefore, the slope of the line parallel to the $x$- axis will be 0 as well.
This is only possible if $A=0$as$m={\displaystyle \frac{-A}{B}}=0\Rightarrow -A=0\Rightarrow A=0$
Substitute$A=0$ in the general equation of the parallel line $Ax+By+C_1=0$
This gives us
$$\begin{gathered} 0x+By+C_1=0 \\ \Rightarrow By=-C_1 \\ \Rightarrow y={\displaystyle \frac{B}{-C_1}}=-{\displaystyle \frac{B}{C_1}} \end{gathered}$$
Let$-{\displaystyle \frac{B}{C_1}}=a$.
Thus the equation becomes $y=a$
Hence the equation of any line parallel to $x$- axis is $y=a$

Note:
 A common mistake made by many students is that they tend to take the coefficient of $x$ in the numerator and that of $y$ in the denominator to calculate the slope. One needs to be careful with this substitution which is the key to answering such questions.