Question

Solve for \[x\] and \[y\]: \[6x + 3y = 6xy\] and \[2x + 4y = 5xy\]

Answer 1

Hint: In the given question, we have been given that there is a pair of equations. For solving this, we are going to reduce the equations to the lowest term if possible. When there are the linear equations with two variables \[\left( {say{\rm{\, m \, and n}}} \right)\] with the terms being multiplied with some constants separately and also a term containing a constant with the product of the two variables \[\left( {mn} \right)\], we first divide the equations by the product of the variables and then solve them like normal equations.

Complete step-by-step answer:
The given system of equations is:
\[6x + 3y = 6xy\] and \[2x + 4y = 5xy\]
Reducing the first equation into the lowest term by dividing by \[3\], we get:
\[2x + y = 2xy\]
Now, dividing both of the equations by the product of the variables, i.e., \[xy\], we have:
\[\dfrac{{2x + y}}{{xy}} = \dfrac{{2xy}}{{xy}}\] and \[\dfrac{{2x + 4y}}{{xy}} = \dfrac{{5xy}}{{xy}}\]
Simplifying the equations, we get,
\[\dfrac{2}{y} + \dfrac{1}{x} = 2\] …(i)
$\Rightarrow$ \[\dfrac{2}{y} + \dfrac{4}{x} = 5\]
Subtracting the two equations, we get:
\[\dfrac{4}{x} - \dfrac{1}{x} = 5 - 2\]
$\Rightarrow$ \[\dfrac{3}{x} = 3\]
Now, solving for \[x\], we have:
\[x = 1\]
Putting \[x = 1\] in (i), we have:
$\Rightarrow$ \[\dfrac{2}{y} + 1 = 2\]
Solving for \[y\],
$\Rightarrow$ \[\dfrac{2}{y} = 1 \Rightarrow y = 2\]
Thus, \[x = 1\] and \[y = 2\]

Note: In this question we divided the two sides of the equation by the product of the variables so as to solve the question effectively. We could have done it in a different way too – equate any one variable of the two equations, subtract the two equations, then just solve the equation by dividing the remaining two expressions.