Question

How do you solve the system of equations $2x-3y=-4$ and $3y-x=5$ ?

Answer 1

Hint: We express the two given equations as functions of a single variable $y$ . Then, we substitute one equation into another to get the value of variable $x$ and then put this value of $x$ in one of the equations to get the value of $y$ . These values are the solution.

Complete step by step answer:
The two linear equations we have are
$2x-3y=-4.....\left( 1 \right)$ and
$3y-x=5.....\left( 2 \right)$
We first multiply $\left( -1 \right)$ to both sides of the equation $\left( 1 \right)$ . The equation thus becomes,
$\Rightarrow 3y-2x=4.....\left( 3 \right)$
We then add $2x$ to both sides of the above equation. The equation thus becomes,
$\Rightarrow 3y=4+2x$
We then divide both sides of the above equation by $3$ . The equation thus becomes,
$\Rightarrow y=\dfrac{4+2x}{3}.....\left( 4 \right)$
Similarly, we add $x$ to both sides of equation $\left( 2 \right)$ . The equation thus becomes,
$\Rightarrow 3y=5+x$
We then divide both sides of the above equation by $3$ . The equation thus becomes,
$\Rightarrow y=\dfrac{5+x}{3}.....\left( 5 \right)$
In the above equations we see that $y$ is written as a function of $x$ . We now take the above equation $\left( 5 \right)$ and substitute the expression of the right-hand side of the above equation in place of $y$ in equation $\left( 4 \right)$ as shown below
$\Rightarrow y=\dfrac{4+2x}{3}$
$\Rightarrow \dfrac{5+x}{3}=\dfrac{4+2x}{3}$
We multiply both sides of the equation by $3$ and get,
$\Rightarrow 5+x=4+2x$
We now add $-x$ to both sides of the above equation and get,
$\Rightarrow 5+x-x=4+2x-x$
This on simplification gives,
$\Rightarrow 5=4+x$
We now subtract $4$ from both sides of the above equation and get,
$\Rightarrow 5-4=x$
This on simplification gives,
$\Rightarrow x=1$
We now substitute this value of $x$ in equation $\left( 5 \right)$ and get,
$\begin{align}
  & \Rightarrow y=\dfrac{5+1}{3} \\
 & \Rightarrow y=2 \\
\end{align}$
Therefore, we can conclude that the solution of the given linear system is $x=1,y=2$ .

Note:
In the given problem, we must be careful while substituting one equation in another and most of the students make mistakes here. This problem can also be solved graphically by taking the two equations as two lines and the point of intersection of these two lines gives the required solution.