Question

Solve the following set of simultaneous equations:
                   \[2x-7y=7;3x+y=22\]

Answer 1

Hint: In the above question, we have two linear equations which are in two variables i.e. x and y. So, if we eliminate one of the two variables, then we can easily get the value of the other variable.
Complete step-by-step answer:
The equations given to us are:
$2x-7y=7$……… (i) and $3x+y=22$……….(ii)
Now, we will eliminate one variable by using the elimination method of solving a set of simultaneous equations.
To eliminate y variable, we can multiply equation (ii) by 7, and then add it to equation (i). On multiplying equation (i) by 7, we get:
$7\times (3x+y)=22\times 7$
$\Rightarrow 21x+7y=154$…….. (iii)
Now, on adding eq. (i) and (iii), we get:
$(2x-7y)+(21x+7y)=7+154$
$\begin{align}
  & \Rightarrow 23x=161 \\
 & \Rightarrow x=\dfrac{161}{23} \\
 & \Rightarrow x=7 \\
\end{align}$
Now, as we have got the value of one variable i.e. x, we can substitute this value of x in any of the above equations and we can find the value of y as well.

So, let us consider the equation (ii), i.e. $3x+y=22$ .
Now, substituting the value x = 7 in the above equation, we get:
$3\times 7+y=22$
\[\Rightarrow 21+y=22\]
$\Rightarrow $ $y=1$
Therefore, the solution of the set of simultaneous equations is $x=7$ , and $y=1$ .

Note: There is also one another method, which one can say is the shortcut to solve the above question. That is known as the substitution method, in that, we can solve the above question in a lesser number of steps. What we have to do, is that just take any of the two given equations, let's say, I take the first one $2x-7y=7$………..(i)
Now, this equation can further be manipulated to get the value of x in terms of y:
$\Rightarrow$ $x=\dfrac{7+7y}{2}$.
Now, we can substitute this value of x in the second equation.
So, this will make that equation, in terms of y, therefore giving us the value of y. Then, we will automatically get the value of x after placing that value of y in the above equation.