Question

How do you solve the systems \[2x+y=4\] and \[6x+3y=12\]?

Answer 1

Hint: In this problem, we have to solve the given system of equations to find the value of x and y. We can divide the second equation by 3, so that we will get the equation which is similar to the first equation. Therefore, we can say that both equations lie on the same line and we can find the intercept of one equation to plot the graph and find its solution.

Complete step by step solution:
We know that the given system of equations to be solved are,
\[2x+y=4\] ……… (1)
\[6x+3y=12\] …….. (2)
We can divide the equation (2) by the number 3, we get
\[\Rightarrow \dfrac{6x}{3}+\dfrac{3y}{3}=\dfrac{12}{3}\]
\[\Rightarrow 2x+y=4\] ………. (3)
Now we can see that the equation (1) and equation (3) are similar and lie on the same line.
Therefore, we can say that the equations have infinitely many solutions.
We can find the intercepts through which both the lines pass through.
We know that for x-intercept the value of y is 0,
\[\begin{align}
  & \Rightarrow 2x+0=4 \\
 & \Rightarrow x=2 \\
\end{align}\]
We also know that for y-intercept the value of x is 0,
\[\Rightarrow y=4\]
Therefore, the x-intercept is \[\left( 2,0 \right)\] and the y-intercept is \[\left( 0,4 \right)\].
Therefore, we can say that the equation passes through the same line at the same intercepts and has infinitely many solutions.
We can plot the graph,

Note: We can also solve the equations in another method.
We know that the given system of equations to be solved are,
\[2x+y=4\] ……… (1)
\[6x+3y=12\] …….. (2)
We can now subtract the equation by elimination method.
We can now multiply 3 to the equation (1), we get
\[\Rightarrow 6x+3y=12\]
\[\Rightarrow 6x+3y-12=0\]…… (3)
 We can now write the equation (2) as,
\[\Rightarrow 6x+3y-12=0\]…… (4)
Now we can subtract the above equations (3) and (4), we get
\[\begin{align}
  & \Rightarrow 6x+3y-12-\left( 6x+3y-12 \right)=0 \\
 & \Rightarrow 6x+3y-12-6x-3y+12=0 \\
\end{align}\]
Now we can cancel similar terms and simplify, we get
\[\Rightarrow 0=0\]
Therefore, the given two systems of equations have infinitely many solutions.