Question

How do you solve the system of linear equations $3x+4y=5$ and $9x+3y=51$?

Answer 1

Hint: For solving the given pair of linear equations, we will use the method of elimination. For this, we have to make the coefficient of one of the variables in both the equations equal. Therefore, we will multiply the first equation $3x+4y=5$ by $3$ so that it will become $9x+12y=15$. Then we will subtract the second equation $9x+3y=51$ from this equation $9x+12y=15$, after which the variable x will get eliminated and we will get the value for the variable y. Finally, on substituting the obtained value of y into any of the given equations, we will obtain the value of the variable x also.

Complete step by step answer:
The system of the linear equations given in the above question is
$\begin{align}
  & \Rightarrow 3x+4y=5........\left( i \right) \\
 & \Rightarrow 9x+3y=51........\left( ii \right) \\
\end{align}$
Let us use the elimination method to eliminate the variable x from the given pair of equations. For this, we multiply the equation (i) by three to get
$\Rightarrow 9x+12y=15........\left( iii \right)$
Now, we subtract the equation (ii) from the equation (ii) to get
\[\begin{align}
  & \Rightarrow 9x+12y-9x-3y=15-51 \\
 & \Rightarrow 12y-3y=-36 \\
 & \Rightarrow 9y=-36 \\
\end{align}\]
Now, we divide both the sides of the above equation by \[9\] to get
$\begin{align}
  & \Rightarrow \dfrac{9y}{9}=\dfrac{-36}{9} \\
 & \Rightarrow y=-4 \\
\end{align}$
Now, we substitute the above equation in the equation (i) to get
$\begin{align}
  & \Rightarrow 9x+3\left( -4 \right)=51 \\
 & \Rightarrow 9x-12=51 \\
\end{align}$
Adding $12$ both the sides, we get
$\begin{align}
  & \Rightarrow 9x-12+12=51+12 \\
 & \Rightarrow 9x=63 \\
\end{align}$
Finally, we divide both the side of the above equation by $9$ to get
\[\begin{align}
  & \Rightarrow \dfrac{9x}{9}=\dfrac{63}{9} \\
 & \Rightarrow x=7 \\
\end{align}\]

Hence, we have solved the given system of linear equations and obtained the solution as \[x=7\] and $y=-4$.

Note: We can also use the method of substitution or cross multiplication for solving the given pair of equations. Also, we must check the obtained values of x and y by substituting them into both the given equations and confirm whether they are satisfying them or not.