Question

Write the condition of linear equation having infinite number of solutions for \[{{\text{a}}_1}{\text{x + }}{{\text{b}}_1}{\text{y + }}{{\text{c}}_1} = {\text{0}}\] and \[{{\text{a}}_2}{\text{x + }}{{\text{b}}_2}{\text{y + }}{{\text{c}}_2} = {\text{0}}\].

Answer 1

Hint:First of all, by reading the question we can understand that we have to write the condition for coincident lines. Because coincident lines have an infinite number of solutions. By following the below given process step by step, you can get a clear solution.

Complete step-by-step answer:
The intersecting points of the two lines are called solutions of the given equations.
If the pair of linear equations have a unique or infinite number of solutions then it is said to be a consistent pair of linear equations.
Given that,
\[{{\text{a}}_1}{\text{x + }}{{\text{b}}_1}{\text{y + }}{{\text{c}}_1} = {\text{0}}\]
\[{{\text{a}}_2}{\text{x + }}{{\text{b}}_2}{\text{y + }}{{\text{c}}_2} = {\text{0}}\]

Condition for infinite solutions:
The equations are consistent and dependent and if should have infinitely many solutions then,
\[\dfrac{{{{\text{a}}_1}}}{{{{\text{a}}_2}}} = \dfrac{{{{\text{b}}_1}}}{{{{\text{b}}_2}}} = \dfrac{{{{\text{c}}_1}}}{{{{\text{c}}_2}}}\]
For having an infinitely many solutions, the linear equations should satisfy some conditions.
1) The linear equations should be coincident lines.
2) They should have the same \[Y\]– intercept.
3) If the system of linear equations have the same \[Y\]– intercept and slope, then the two lines are actually in the exact same line.
So, we can say that, if the given two lines are the same line, it should have infinitely many solutions.

Note:Do not get confused between consistent and inconsistent. If the system of lines has at least one solution, then we can say them as consistent. Coincident lines which have infinite number of solutions and intersecting lines which have one solution obey the above condition. If the system of lines does not have even one solution, then it is said to be inconsistent. Parallel lines which do not have even one solution obey this condition. Parallel lines have zero solutions because they do not intersect at even one point.