Question

How do you solve the systems \[3x+2y=2\] and \[-2x+y=8\]?

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 multiply 2 to the first equation and 3 to the second equation. We can then use elimination methods to cancel similar terms to get one of the unknown variable values, which we can substitute in one equation to get the other unknown variable value.

Complete step-by-step solution:
We know that the given system of equations to be solved are,
\[3x+2y=2\] ……… (1)
\[-2x+y=8\] …….. (2)
We can now add 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 2 to the equation (1), we get
\[\Rightarrow 6x+4y=4\]
\[\Rightarrow 6x+4y-4=0\]…… (3)
 We can now multiply 3 to the equation (2), we get
\[\Rightarrow -6x+3y=24\]
\[\Rightarrow -6x+3y-24=0\]…… (4)
Now we can add the above equations (3) and (4), we get
\[\begin{align}
  & \Rightarrow 6x+4y-4+\left( -6x+3y-24 \right)=0 \\
 & \Rightarrow 6x+4y-4-6x+3y-24=0 \\
\end{align}\]
Now we can cancel similar terms and simplify, we get
\[\begin{align}
  & \Rightarrow 4y+3y-4-24=0 \\
 & \Rightarrow 7y-28=0 \\
 & \Rightarrow y=4 \\
\end{align}\]
Therefore, the value of y is 4.
Now we can substitute the y value in equation (1), we get
\[\begin{align}
  & \Rightarrow 3x+2\left( 4 \right)=2 \\
 & \Rightarrow 3x=-6 \\
 & \Rightarrow x=-2 \\
\end{align}\]
Therefore, the value of x = -2 and y = 4.


Note: Students make mistakes while multiplying the correct number to the equations for the similar terms to be cancelled. We should concentrate while using elimination methods to add/subtract the equations to get one of the values and to substitute it to get the other value.