Question

How do you solve for x in the equation \[xy = x + y\] ?

Answer 1

Hint:Here we need to solve for \[x\], in which the equation consists of two variables \[x\] and \[y\], hence to solve for both \[x\] and \[y\] we can use substitution method to get both the values, but we need to solve only for \[x\], hence we can solve by taking \[x\] common from both LHS and RHS part we can find the solution for \[x\].

Complete step by step answer:
The given equation is
\[xy = x + y\] .…………………. 1
which is of the form LHS (Left Hand Side) = RHS (Right Hand Side) in which LHS is \[xy\] and RHS is \[x + y\].Here we need to solve for \[x\].In equation 1 we can see that \[x\] is present in both LHS and RHS of the given equation. Hence for this first we need to transpose \[x\] from the RHS part to LHS of the equation.After transposing the terms of \[x\] to LHS side we get
\[xy - x = y\] ……………………… 2

We can see in equation 2 is that the \[x\] term is common in LHS side, hence by taking \[x\] common the equation becomes
\[x\left( {y - 1} \right) = y\] ………………………. 3
Hence to solve for \[x\] we need to divide \[\left( {y - 1} \right)\] on both sides in equation 3.
By dividing the terms, we get
\[\dfrac{{x(y - 1)}}{{y - 1}} = \dfrac{y}{{y - 1}}\]
Where \[\left( {y - 1} \right)\] divides by one in the LHS part and RHS part remains the same as there is no common term which divides.

Hence, by solving this we got the \[x\] value as \[x = \dfrac{y}{{y - 1}}\].

Note:To solve both $x$ and $y$ use a substitution method to find the values and to solve for one variable in the equation first we need to find the common term from the equation which they have asked and next transposing the terms based on LHS and RHS side. If they ask to solve for $y$ then we have to take $y$ as a common term and proceed to solve.