Question

How do you solve $3y=14+x$ and $x+22=5y$?

Answer 1

Hint: In the above question, we are given two linear equations in the two variables. So we can solve these using the method of elimination. We see that the coefficient of x is the same in both of the given equations. So on subtracting the equations, we will obtain a linear equation in terms of y only which can be easily solved for obtaining the value of y. Finally, on substituting the obtained value of y into any of the given equations, we will get the value of x also.

Complete step-by-step solution:
The first equation in the above question are given as
$\begin{align}
  & \Rightarrow 3y=14+x \\
 & \Rightarrow x+14=3y........\left( i \right) \\
\end{align}$
And the second equation is
$\Rightarrow x+22=5y.......\left( ii \right)$
We observe in the equations (i) and (ii) that the coefficient of x in both of these is the same and is equal to one in both. So we can subtract the two so as to vanish the variable x and get an equation in terms of the variable y along. Therefore we subtract the equation (i) from the equation (ii) to get
$\begin{align}
  & \Rightarrow x+22-\left( x+14 \right)=5y-3y \\
 & \Rightarrow x+22-x-14=2y \\
 & \Rightarrow 8=2y \\
\end{align}$
Dividing both the sides of the above equation by $2$, we get
$\begin{align}
  & \Rightarrow \dfrac{8}{2}=\dfrac{2y}{2} \\
 & \Rightarrow 4=y \\
 & \Rightarrow y=4 \\
\end{align}$
Now, we substitute this in the equation (i) to get
$\begin{align}
  & \Rightarrow x+22=5\left( 4 \right) \\
 & \Rightarrow x+22=20 \\
\end{align}$
Finally, on subtracting $22$ from both sides of the above equation we get
$\begin{align}
  & \Rightarrow x+22-22=20-22 \\
 & \Rightarrow x=-2 \\
\end{align}$
Hence, the solution of the given system of equations is $x=-2$ and $y=4$.

Note: We have substituted the obtained value of the variable y into the first equation $3y=14+x$ to get the value of x. This means that the obtained solution will definitely satisfy the first equation. But to check the correctness of the solution, we must substitute it into the second equation $x+22=5y$ and confirm whether it satisfies it too or not. If it does not satisfy the second equation, then this means that our obtained solution is incorrect.