Question

How do you solve the system $4x-3y=34,3x+9y=-42$?

Answer 1

Hint: To solve the given system we will use the elimination method. First we will multiply the first equation by 3 to make the coefficients of y variable in both equations same. Then add both the equations to get the value of x. Then substitute the obtained value of x in the first equation to get the value of y.

Complete step-by-step solution:
We have been given the system $4x-3y=34,3x+9y=-42$.
We have to solve the given system of equations.
We have
$4x-3y=34.......(i)$
$3x+9y=-42......(ii)$
First we will multiply equation (i) by 3, then we will get
$\begin{align}
  & \Rightarrow 4x\times 3-3y\times 3=34\times 3 \\
 & \Rightarrow 12x-9y=102...........(iii) \\
\end{align}$
Now, adding equation (ii) and equation (ii) we will get
$\begin{align}
  & \underline{\begin{align}
  & 12x-9y=102 \\
 & 3x+9y=-42 \\
\end{align}} \\
 & 15x=60 \\
\end{align}$
Now, simplifying the above obtained equation we will get
$\begin{align}
  & \Rightarrow x=\dfrac{60}{15} \\
 & \Rightarrow x=4 \\
\end{align}$
Now, substituting the value of x in equation (i) we will get
$\Rightarrow 4\times 4-3y=34$
Now, simplifying the above obtained equation we will get
$\begin{align}
  & \Rightarrow 16-3y=34 \\
 & \Rightarrow -3y=34-16 \\
 & \Rightarrow -3y=18 \\
 & \Rightarrow y=-\dfrac{18}{3} \\
 & \Rightarrow y=-6 \\
\end{align}$
Hence on solving the given system we get the values of x and y as 4 and $-6$ respectively.

Note: Using the elimination method is the easiest way to solve the system. The possibility of mistake is that if students try to directly add or subtract the given equation without analyzing the coefficients of the variable they will get the result as
$\begin{align}
  & \underline{\begin{align}
  & 4x-3y=34 \\
 & 3x+9y=-42 \\
\end{align}} \\
 & x+3y=-8 \\
\end{align}$
We again get an equation with two variables. So it is necessary to make coefficients of one variable equal to eliminate the variable from the equation.