Question

How many solutions does the following system of equations have \[3y-6x=9,2y-4x=6\]?

Answer 1

Hint: In this problem, we have to find how many solutions are there in the given system of equations. We can first divide the both equations by similar or multiple numbers in order to simplify the given system of equations. we will get two identical equations and the both equations denote the same line.

Complete step by step answer:
We know that the given system of equations is,
\[\begin{align}
  & 3y-6x=9.....(1) \\
 & 2y-4x=6.....(2) \\
\end{align}\]
We can take the equation (1) and divide each term by 3, we get
\[\Rightarrow y-2x=3\] …. (3)
We can take equation (2) and divide each term by 2, we get
\[\Rightarrow y-2x=3\] …. (4)
Now we can see that the equation (3) and equation (4) are identical and actually denotes the same line.
Hence for any \[x={{x}_{1}}\] we can get \[{{y}_{1}}=2{{x}_{1}}+3\] and \[\left( {{x}_{1}},{{y}_{1}} \right)\] is a solution to the equation and we can have infinite \[\left( {{x}_{1}},{{y}_{1}} \right)\] all lying on the line \[y=2x+3\].

Therefore, there are infinite solutions for the given system of equations \[3y-6x=9,2y-4x=6\].

Note: Students make mistakes while finding for the identical equations, which will decide the number of solutions. We should know that, to solve these types of problems, we have to understand the concept of consistency and inconsistent linear systems. We can first divide the both equations by similar or multiple numbers in order to simplify the given system of equations to check whether the equations are identical to find the number of solutions.