Question

How do you solve $y+3x=11$ and $2y-x=2$?

Answer 1

Hint: We have a system of two linear equations. We will use the Gauss elimination method to solve this system of equations. We will multiply the second equation by 3. Then we will add this equation to the first equation and eliminate the $x$ variable. We will find the value of $y$ and then substitute it in one equation to obtain the value of $x$.

Complete step by step answer:
The given system of linear equations is the following,
$y+3x=11....(i)$
$2y-x=2....(ii)$
To solve this system of linear equations, we will use the Gauss elimination method. According to this method, we will first make the coefficient of one of the variables same. Then we will eliminate this variable using appropriate binary operation.
Let us multiply equation $(ii)$ by 3. So, we will get the following,
$6y-3x=6....(iii)$
Now, we can see that the coefficient of the variable $x$ is the same in equation $(i)$ and equation $(iii)$. Next, we will eliminate the variable $x$. To do this, let us add equation $(i)$ and equation $(iii)$. So, we have the following,
$\begin{matrix}
   {} & y+3x=11 \\
   + & 6y-3x=6 \\
   {} & 7y=17 \\
\end{matrix}$
So, we get the value $y=\dfrac{17}{7}$. Now, we will substitute this value of $y$ in equation $(i)$. So, we have the following,
$\dfrac{17}{7}+3x=11$
Solving the above equation for $x$, we get
$\begin{align}
  & 3x=11-\dfrac{17}{7} \\
 & \Rightarrow 3x=\dfrac{77-17}{7} \\
 & \Rightarrow 3x=\dfrac{60}{7} \\
 & \therefore x=\dfrac{20}{7} \\
\end{align}$
Therefore, we obtain the solution of the system of linear equations as $x=\dfrac{20}{7}$ and $y=\dfrac{17}{7}$.

Note:
There are other methods to solve a system of linear equations. These are methods of substitution and method of graphing. In the method of graphing, we draw the straight lines represented by the given equations and the solution is the point of their intersection. In the method of substitution, we substitute one variable in terms of the other and find its value. Then we substitute this value in one of the equations to find the value of the other variable.