Question

How do you solve the systems \[3x+4y=2\] and \[2x+3y=1\]?

Answer 1

Hint: In this problem, we have to find the solution and find the value of x and y from the given system of equations. We can first multiply the number 3 to the first equation and the number 2 to the second equation in order to get similar terms in which one of the variables can be cancelled while using elimination method. We will get the value of x or y, we can substitute the result value to get the value of another variable.

Complete step by step solution:
We know that the given system of equations to be solved are,
\[2x+3y=1\] ……… (1)
\[3x+4y=2\] …….. (2)
We can now subtract the equation by elimination method.
We should know that to solve by elimination method, we should have similar terms to be cancelled, so we can multiply both equations with numbers to get similar terms.
We can now multiply 3 to the equation (1), we get
\[\Rightarrow 6x+9y=3\]
\[\Rightarrow 6x+9y-3=0\]…… (3)
 We can now multiply 2 to the equation (2), we get
\[\Rightarrow 6x+8y=4\]
\[\Rightarrow 6x+8y-4=0\]…… (4)
Now we can subtract the above equations (3) and (4), we get
\[\begin{align}
  & \Rightarrow 6x+9y-3-\left( 6x+8y-4 \right)=0 \\
 & \Rightarrow 6x+9y-3-6x-8y+4=0 \\
\end{align}\]
Now we can cancel similar terms and simplify, we get
\[\begin{align}
  & \Rightarrow 9y-8y-3+4=0 \\
 & \Rightarrow y=-1 \\
\end{align}\]
Therefore, the value of y is -1.
Now we can substitute the y value in equation (1), we get
\[\begin{align}
  & \Rightarrow 2x+3\left( -1 \right)=1 \\
 & \Rightarrow 2x=4 \\
 & \Rightarrow x=2 \\
\end{align}\]

Therefore, the value of x = 2 and y = -1.

Note: Students make mistakes while multiplying the correct number to the equations for the similar terms to be cancelled. We can also directly find the value by substituting the one equation into the other. We can add/subtract the equations by elimination method by multiplying numbers to the equation in order to cancel the similar terms. We can substitute the values found in any of the equations, to verify and check for the correct answer.
When x = 2 and y = -1 in \[2x+3y=1\]
\[\begin{align}
  & \Rightarrow 2\left( 2 \right)+3\left( -1 \right)=1 \\
 & \Rightarrow 4-3=1 \\
\end{align}\]
Therefore, the values x = 2 and y = -1 are correct.