Question

Which of the following represent a line parallel to the \[x\]–axis?
(a) \[x + y = 3\] (b) \[2x + 3 = 7\]
(c) \[2 - y - 3 = y + 1\] (d) \[x + 3 = 0\]

Answer 1

Hint:
Here, we need to check which of the given lines is parallel to the \[x\]–axis. We will use the general form of a line parallel to the \[x\]–axis or the \[y\]–axis to check which of the given lines is parallel to the \[x\]–axis and find the correct option.

Complete step by step solution:
We will check each of the options one by one to find which of the given lines are parallel to the \[x\]–axis.
A line parallel to the \[x\]–axis is of the form \[y + a = 0\], where \[a\] is not equal to 0.
The first equation is \[x + y = 3\].
Rewriting the equation \[y + a = 0\], we get
\[0 \cdot x + 1 \cdot y + a = 0\]
The linear equations in two variables \[{a_1}x + {b_1}y + {c_1} = 0\] and \[{a_2}x + {b_2}y + {c_2} = 0\] have unique solution if \[\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}\].
The linear equations in two variables \[{a_1}x + {b_1}y + {c_1} = 0\] and \[{a_2}x + {b_2}y + {c_2} = 0\] have infinitely many solutions if \[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}\].
The linear equations in two variables \[{a_1}x + {b_1}y + {c_1} = 0\] and \[{a_2}x + {b_2}y + {c_2} = 0\] have no solution if \[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}\], and thus, are parallel.
Comparing \[0 \cdot x + 1 \cdot y + a = 0\] to the standard form \[{a_1}x + {b_1}y + {c_1} = 0\], we get
\[{a_1} = 0\], \[{b_1} = 1\], and \[{c_1} = a\]
Comparing \[x + y = 3\] to the standard form \[{a_2}x + {b_2}y + {c_2} = 0\], we get
\[{a_2} = 1\], \[{b_2} = 1\], and \[{c_2} = - 3\]
Now, we will find the ratios of the coefficients of \[x\], \[y\], and the constant.
Dividing \[{a_1} = 0\] by \[{a_2} = 1\], we get
\[\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{0}{1} = 0\]
Dividing \[{b_1} = 1\] by \[{b_2} = 1\], we get
\[\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{1}{1} = 1\]
Therefore, we can observe that \[\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}\].
Therefore, the lines \[0 \cdot x + 1 \cdot y + a = 0\] and \[x + y = 3\] have a unique solution.
This means that these lines are intersecting lines.
Hence, \[x + y = 3\] is not parallel to the \[x\]–axis.
Thus, option (a) is incorrect.
The second equation is \[2x + 3 = 7\].
Subtracting 3 from both sides, we get
\[\begin{array}{l} \Rightarrow 2x + 3 - 3 = 7 - 3\\ \Rightarrow 2x = 4\end{array}\]
Dividing both sides by 2, we get
\[ \Rightarrow x = 2\]
Rewriting the equation, we get
\[ \Rightarrow x - 2 = 0\]
We can observe that this equation is of the form \[x + a = 0\].
A line parallel to the \[y\]–axis is of the form \[x + a = 0\], where \[a\] is not equal to 0.
Thus, the line \[2x + 3 = 7\] is parallel to the \[y\]–axis.
Therefore, the line \[2x + 3 = 7\] is not parallel to the \[x\]–axis.
Thus, option (b) is incorrect.
The third equation is \[2 - y - 3 = y + 1\].
Adding and subtracting the like terms, we get
\[ \Rightarrow - 1 - y = y + 1\]
Rewriting the equation, we get
\[ \Rightarrow y + y = - 1 - 1\]
Simplifying the expression, we get
\[ \Rightarrow 2y = - 2\]
Dividing both sides by 2, we get
\[ \Rightarrow y = - 1\]
Rewriting the equation, we get
\[ \Rightarrow y + 1 = 0\]
We can observe that this equation is of the form \[y + a = 0\].
A line parallel to the \[x\]–axis is of the form \[y + a = 0\], where \[a\] is not equal to 0.
Therefore, the line \[2 - y - 3 = y + 1\] is parallel to the \[x\]–axis.
Thus, option (c) is the correct option.
The fourth equation is \[x + 3 = 0\].
We can observe that this equation is of the form \[x + a = 0\].
A line parallel to the \[y\]–axis is of the form \[x + a = 0\], where \[a\] is not equal to 0.
Thus, the line \[x + 3 = 0\] is parallel to the \[y\]–axis.
Therefore, the line \[x + 3 = 0\] is not parallel to the \[x\]–axis.

Note:
A line parallel to the \[x\]–axis is of the form \[y + a = 0\], where \[a\] is not equal to 0. If \[a\] is equal to 0, then the line becomes \[y = 0\], which is coincident with the \[x\]–axis, and not parallel. Hence, \[a\] cannot be equal to 0. Similarly, a line parallel to the \[y\]–axis is of the form \[x + a = 0\], where \[a\] is not equal to 0.