Question

Solve the following:
\[9x + 3y + 12 = 0\]
$18x + 6y + 24 = 0$

Answer 1

Hint: This is a question of simultaneous equations. In these types of questions first we have to find the value of one variable in terms of another variable and then we put this value in the second equation to get the desired value of variables. This is one method called the substitution method of equations in two variables. There are other methods of doing this question but this is the easiest one. But there is one twist in this question which you will get to know when you will try to solve this question.

Complete step by step answer:
In the given question, we have
We can represent an equation of line in a standard format as $ax + by + c = 0$.
We have,
\[9x + 3y + 12 = 0 - - - - - - \left( 1 \right)\]
$18x + 6y + 24 = 0 - - - - - - \left( 2 \right)$
Now, we can write ${a_1} = 9\,,\,{b_1} = 3\,,\,{c_1} = 12$
${a_2} = 18\,,\,{b_2} = 6\,,\,{c_2} = 24$
Now,
$ \Rightarrow \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{9}{{18}} = \dfrac{1}{2}\,;\,\,\,\,\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{3}{6} = \dfrac{1}{2}\,;\,\,\,\,\dfrac{{{c_1}}}{{{c_2}}} = \dfrac{{12}}{{24}} = \dfrac{1}{2}$
Therefore, we can say that from the above result
$ \Rightarrow \dfrac{{{a_1}}}{{{a_2}}} = \,\,\dfrac{{{b_1}}}{{{b_2}}} = \,\,\dfrac{{{c_1}}}{{{c_2}}}$
So from the above condition we know that both lines are coincident lines or we can say that both equations represent the same line. So we will get infinite solutions for this pair of linear equations.

Note:
 It is the system of concurrent lines, which are in the form of straight lines. The highest degree of the linear equation in two variables is one. The general form of linear equation in two variables is $ax + by + c = 0$. The pair of linear equations in two variables are used to solve the equations by using different methods, such as graphical, elimination, substitution and cross-multiplication methods. Pair of linear equations in two variables are denoted graphically or geometrically in the form of straight lines. There are mainly three types of straight lines that are possible. They are Intersecting lines, Parallel Lines and Coincident lines.