Question

how do you solve \[x=\dfrac{y-1}{y+1}\] for y?

Answer 1

Hint: Any equation can be solved by taking one variable to one side and another variable to the other side of the equation. one side must be solved step-by-step to get through the solution. We can use the distributive property and do the addition, subtraction, multiplication, and division operations wherever necessary in such a way to simplify the equation.

Complete step by step answer:
As per the given question we need to find the value of y from the given expression.
Firstly we multiply the expression with \[y+1\] on both sides. Then the equation becomes
\[\Rightarrow \]\[x\times \left( y+1 \right)=\dfrac{y-1}{y+1}\times \left( y+1 \right)\]
On expanding the multiplication on the left-hand side and on the right-hand side \[y+1\] is common in both numerator and denominator. So we can cancel it.
 Then the equation becomes
\[\Rightarrow \]\[xy+x=y-1\]
Now we add \[-xy\] on both sides. Then the equation becomes
\[\begin{align}
  & \Rightarrow xy+x-xy=y-1-xy \\
 & \Rightarrow x=y-1-xy \\
\end{align}\]
In the right hand side of the equation we can take y common in the terms \[y,-xy\]then the equation becomes
\[\Rightarrow \]\[x=y\left( 1-x \right)-1\]
Now we add 1 on both sides then the equation becomes
\[\begin{align}
  & \Rightarrow x+1=y\left( 1-x \right)-1+1 \\
 & \Rightarrow x+1=y\left( 1-x \right)+0 \\
\end{align}\]
Now we divide with \[\left( 1-x \right)\] on both sides. Then the equation becomes
\[\Rightarrow \]\[\dfrac{x+1}{1-x}=y\dfrac{1-x}{1-x}\]
Here we have both numerator and denominator. So we can cancel it. Then the equation becomes
\[\Rightarrow \]\[\dfrac{x+1}{1-x}=y\]
Therefore, the value of y is \[\dfrac{x+1}{1-x}\].

Note:
 In order to solve these types of problems, we need to have knowledge of the basic arithmetic functions. We should know the distributive property, additive property, associative property, commutative property to solve these types of problems. We should avoid calculation mistakes to get the correct solution.