Question

Graphically, the pair of equations $6x-3y+10=0$, $2x-y+9=0$ represents two lines which are
(a) intersecting at exactly one point
(b) intersecting at exactly two points
(c) coincident
(d) parallel

Answer 1

Hint: We are given a pair of linear equations in two variables. We will compare the coefficients of the given two equations. The relation between the coefficients tells us about the nature of the solution to the two equations. It also tells us the nature of the lines represented by the two equations.

Complete step by step answer:
Let ${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0$ and ${{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0$ be two general linear equations. The conditions for consistency of a system of equations is as follows:
(i) If $\dfrac{{{a}_{1}}}{{{a}_{2}}}\ne \dfrac{{{b}_{1}}}{{{b}_{2}}}$, then there is a unique solution;
(ii) If $\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}\ne \dfrac{{{c}_{1}}}{{{c}_{2}}}$, then there is no solution; and
(iii) If $\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}=\dfrac{{{c}_{1}}}{{{c}_{2}}}$, then there are infinite solutions.
From these conditions, we can interpret that if a system of equations satisfies condition (i), then the lines intersect at exactly one point. If the system of equations satisfies condition (ii), then the lines are parallel. And if the system of equations satisfies condition (iii), then the lines are coincident.
The given linear equations are $6x-3y+10=0$ and $2x-y+9=0$. Comparing these two equations with the general equations, we get ${{a}_{1}}=6\text{, }{{b}_{1}}=-3\text{, }{{c}_{1}}=10$ and ${{a}_{2}}=2\text{, }{{b}_{2}}=-1,\text{ }{{c}_{2}}=9$.
Taking the ratios of the corresponding coefficients, we have $\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{6}{2}=\dfrac{3}{1}$, $\dfrac{{{b}_{1}}}{{{b}_{2}}}=\dfrac{-3}{-1}=\dfrac{3}{1}$ and $\dfrac{{{c}_{1}}}{{{c}_{2}}}=\dfrac{10}{9}$. This implies that the given equations satisfy condition (ii) mentioned above. Therefore, we can conclude that the lines represented by the given two equations are parallel.

So, the correct answer is “Option D”.

Note: We can solve this question by other methods. We can plot the graphs of the given linear equations and see that the lines are parallel as shown in the figure below,
 

It is useful to know the conditions for consistency of a system of linear equations as it will save time while solving this type of question.