Question

Do the following pairs represent a pair of coincident lines? Justify your answer .
\[x + \dfrac{1}{7}y = 37\] and \[x + 3y = 7\] .
A.No
B.Yes
C.Cannot determined
D.None of these.

Answer 1

Hint: Definition of coincident lines: Two lines that lie exactly on the top of each other are called coincident lines. Such lines have all points in common and intersect at infinitely many points.

Complete step-by-step answer:
In this question we are given equations of two lines namely \[x + \dfrac{1}{7}y = 37 \] and \[x + 3y = 7\], and we need to check whether these lines are coincident or not To check the coincidence of the given lines, we will check their intersecting points. If the lines have only one point in common, then they are not coincident lines, they are just called intersecting lines. If these lines have infinitely many points in common or after solving them we end up with an obvious result then the lines are coincident lines. So we will solve the given lines and find their coincidence.
The given lines are
\[x + \dfrac{1}{7}y = 37 \] \[\left( 1 \right)\]
\[x + 3y = 7\] \[\left( 2 \right)\]
Subtracting \[\left( 1 \right)\] from\[\left( 2 \right)\]we get,
\[\begin{array}{l}
3y - \dfrac{1}{7}y\, = \, - 30\\
 \Rightarrow \left( {\dfrac{{21 - 1}}{7}} \right)y = - 30\\
 \Rightarrow \dfrac{{20}}{7}y = - 30\\
 \Rightarrow y = \dfrac{{ - 30 \times 7}}{{20}}\\
 \Rightarrow y = \dfrac{{ - 21}}{2}
\end{array}\]
Now , \[y = - \dfrac{{21}}{2}\] \[\left( 3 \right)\]
Put \[\left( 3 \right)\]in\[\left( 2 \right)\], we get
  \[x = \,\dfrac{{77}}{2}\] ………..\[\left( 4 \right)\]
So the only point common in the given lines is \[\left( {\dfrac{{77}}{2}, - \dfrac{{21}}{2}} \right)\] .
We get only one point in common, so the given lines are not coincident lines.
So, the correct answer is “Option A”.

Note: Coincident lines are not the same as that of parallel lines, parallel lines have the same slope, but they never meet, whereas the coincident lines meet at infinitely many points.
Alternate method : The two lines are coincident if they have the same slopes and same intercepts.
Now the general equation of a line is \[y = mx + c\]. Where m is the slope and c is the y-intercept
Comparing the given equations with the general equation of line,
For line \[\left( 1 \right)\] slope =\[ - 7\] and y-intercept =\[259\]
For line\[\left( 2 \right)\]slope= \[\dfrac{{ - 1}}{3}\] and y-intercept =\[\dfrac{7}{3}\]
So we see that both the slopes and intercepts are not equal, so the given lines are not coincident.