Question

How do you solve $2x+7y=10$ and $x-2y=15$?

Answer 1

Hint: Try to find the value of ‘x’ in terms of ‘y’ from the first equation. Then put that value of ‘x’ in the second equation to get the value of ‘y’. Again putting that ‘y’ value in any one of the equations, ‘x’ value can be obtained.

Complete step by step solution:
Solving the simultaneous equations: First we have to take any one of the two equations. Then we have to find the value of ‘x’ in terms of ‘y’. Putting that value of ‘x’ in the second equation we can get the value of ‘y’. Again putting that value of ‘y’ in any of the equations we can get the value of ‘x’.
Considering the first equation
$2x+7y=10$
From here ‘x’ can be obtained in terms of ‘y’ as
$\begin{align}
  & 2x=10-7y \\
 & \Rightarrow x=\dfrac{10-7y}{2} \\
\end{align}$
Now considering the second equation $x-2y=15$
It can be written as
$2y=x-15$
Putting the value of ‘x’ we got earlier in the above equation, we get
$\begin{align}
  & 2y=\dfrac{10-7y}{2}-15 \\
 & \Rightarrow 2y=\dfrac{10-7y-30}{2} \\
 & \Rightarrow 2y\times 2=10-30-7y \\
 & \Rightarrow 4y+7y=-20 \\
 & \Rightarrow 11y=-20 \\
 & \Rightarrow y=-\dfrac{20}{11} \\
\end{align}$
Putting the value of ‘y’ in the second equation, we get
$\begin{align}
  & x-2y=15 \\
 & \Rightarrow x-2\times \left( -\dfrac{20}{11} \right)=15 \\
 & \Rightarrow x-\left( -\dfrac{40}{11} \right)=15 \\
 & \Rightarrow x+\dfrac{40}{11}=15 \\
 & \Rightarrow x=15-\dfrac{40}{11} \\
 & \Rightarrow x=\dfrac{165-40}{11} \\
 & \Rightarrow x=\dfrac{125}{11} \\
\end{align}$
Hence the solution of the system of linear equations $2x+7y=10$ and $x-2y=15$ is $\left( x,y \right)=\left( -\dfrac{20}{11},\dfrac{125}{11} \right)$.
This is the required solution.

Note: There is an alternative method for solving two simultaneous equations. First we have to multiply the equations with required constants to make the coefficient of one variable equal. Then we have to add or subtract the new equations which are formed by multiplying the constants. From there we can get the value of one variable. For other one, put the value of the already obtained variable in one of the equations.