Question

If a pair of linear equation is consistent, then the lines will be:
a)Parallel
b)Always coincident
c)Intersecting or coincident
d)Always intersecting

Answer 1

Hint: Check for the conditions of unique solutions and infinity solutions if pair of straight lines is in form of

${{a}_{1}}x+{{b}_{1}}y={{c}_{1}}$

${{a}_{2}}x+{{b}_{2}}y={{c}_{2}}$

In the question, we are given that if a pair of linear equations is consistent then we have to point out the type of lines it is from the options given.

Before concluding about the answer or which option is correct, we will tell or explain about the term consistent. Lines which intersect at a point which represents the unique solution of the system in two variables. Generally, system of pair of straight lines are represented by ${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0$ and ${{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0$.

In other languages the pair of lines will coincide either or intersect at one point, then the pair of lines will be consistent. Otherwise, they will be considered as consistent.

We check the pair of lines coincide or not using the condition that,

$\dfrac{{{a}_{1}}}{{{b}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}=\dfrac{{{c}_{1}}}{{{c}_{2}}}$

If the above condition satisfies then the lines are considered as consistent as it means the pair of lines are coincident.

There is also another condition that is represented as

$\dfrac{{{a}_{1}}}{{{b}_{2}}}\ne \dfrac{{{b}_{1}}}{{{b}_{2}}}$

If the above condition satisfies then the lines are considered as consistent as it means the pair of lines are intersecting.

So, the correct option is ‘C’.


Note: If the pair of straight lines are parallel then, the lines will be considered as inconsistent their condition will be,

$\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}\ne \dfrac{{{c}_{1}}}{{{c}_{2}}}$