Question

Solve for x and y the given system of linear equations $6x+3y=6xy;2x+4y=5xy$

Answer 1

Hint: There are two unknowns $x$ and $y$ and also two equations to solve. We solve the equations equating the coefficients of one variable and omitting the variable. The other variable remains with the constants. Using the binary operation, we find the value of the other variable. First, we are applying the process of reduction and then the substitution.

Complete step-by-step solution:
The given equations $6x+3y=6xy$ and $2x+4y=5xy$ are linear equations of two variables.
We know that the number of equations has to be equal to the number of unknowns to solve them.
We take the equations as $6x+3y=6xy.....(i)$ and $2x+4y=5xy......(ii)$.
We multiply 3 to the both sides of the second equation and get
$\begin{align}
  & 3\times \left( 2x+4y \right)=3\times 5xy \\
 & \Rightarrow 6x+12y=15xy \\
\end{align}$
We take the equation as $6x+12y=15xy.....(iii)$.
Now we subtract the equation (i) from equation (iii) and get
$\left( 6x+12y \right)-\left( 6x+3y \right)=15xy-6xy$.
We take the similar variables together.
Simplifying the equation, we get
$\begin{align}
  & \left( 6x+12y \right)-\left( 6x+3y \right)=15xy-6xy \\
 & \Rightarrow 9y=9xy \\
 & \Rightarrow 9y\left( 1-x \right)=0 \\
\end{align}$
The possible solutions are $y=0$ or $x=1$.
In the case of $y=0$, we get $x=0$. In the case of $x=1$, we get $y=2$.
Therefore, the values are $x=0,y=0$ or $x=1,y=2$.

Note: We can also find the value of $xy$ or equate that for both equations to find the ratio of the two variables. We use that relation to find the solution for the variables. We multiply the second equation with 3 to equate them and find the ratio.