Question

Solve the following system of equations:
$\begin{align}
  & 2x+3y+8=0 \\
 & 4x+5y+14=0 \\
\end{align}$

Answer 1

Hint: It is given as we have to solve the given system of simultaneous equations by elimination method so multiply the first equation by 2 and multiply the second equation by 1 then subtract the second equation from the first. After subtraction, you will find that x is eliminated and you will be left with y and then simplify and get the value of y and then substitute this value of y in the first equation. It will give you the value of y.

Complete step-by-step solution -
The two simultaneous equations given in the question are:
$\begin{align}
  & 2x+3y+8=0.......Eq.(1) \\
 & 4x+5y+14=0......Eq.(2) \\
\end{align}$
We are going to solve the above simultaneous equations by elimination method in which we transform our equations by multiplying eq. (1) by 2 and then subtract eq.(2) from this new eq.(1) and then x will be eliminated and we get the value of y.
Multiplying eq. (1) by 2 and multiplying eq. (2) by 1 and then subtract eq. (2) from eq. (1) we get,
$\begin{align}
  & \left( 2x+3y+8=0 \right)\times 2 \\
 & \dfrac{-\left( 4x+5y+14=0 \right)\times 1}{\left( 6-5 \right)y+16-14=0} \\
\end{align}$
Simplifying the above equation we get,
$\begin{align}
  & y+2=0 \\
 & \Rightarrow y=-2 \\
\end{align}$
Plugging this value of x in eq. (1) we get,
$\begin{align}
  & 2x+3\left( -2 \right)+8=0 \\
 & \Rightarrow 2x-6+8=0 \\
 & \Rightarrow 2x+2=0 \\
 & \Rightarrow x+1=0 \\
 & \Rightarrow x=-1 \\
\end{align}$
From the above, the solution of system of equations is:
$\begin{align}
  & x=-1; \\
 & y=-2 \\
\end{align}$.

Note: You can verify the solutions that you are getting are correct or not by satisfying the values of x and y in the given system of equations.
$\begin{align}
  & 2x+3y+8=0 \\
 & 4x+5y+14=0 \\
\end{align}$
The values of x and y that we have got above are:
$\begin{align}
  & x=-1; \\
 & y=-2 \\
\end{align}$
Take any one of the two simultaneous equations and substitute the values of x and y in that equation.
$\begin{align}
  & 2x+3y+8=0 \\
 & \Rightarrow 2\left( -1 \right)+3\left( -2 \right)+8=0 \\
 & \Rightarrow -2-6+8=0 \\
 & \Rightarrow -8+8=0 \\
 & \Rightarrow 0=0 \\
\end{align}$
From the above calculations, we have shown that the obtained values of x and y satisfies one of the simultaneous equations i.e. $2x+3y+8=0$.
Hence, we have verified that the values of x and y are correct.
You can check for yourself that the values of x and y that we have got above is satisfying the other equation of the system of equations.