Question

How do you solve for x in $\dfrac{1}{x} = \dfrac{1}{y} + \dfrac{1}{z}$ ?

Answer 1

Hint: Multiplying or dividing the same quantity or number from both sides of an equation gives an equation that’s equivalent to the original equation. Using this idea, we can isolate the variable we require, (x in this case) from the rest of the equation and then simplify further to solve the equation.

Complete Step by Step Solution:
The given question is $\dfrac{1}{x} = \dfrac{1}{y} + \dfrac{1}{z}$ . In order to solve for x, we have to isolate the variable, x from y and z.
Multiplying throughout by xyz,
$\Rightarrow \dfrac{{xyz}}{x} = \dfrac{{xyz}}{y} + \dfrac{{xyz}}{z}$
For each of the terms in the above expression, we can cancel out the common variables in the numerator and denominator, i.e.,
$\Rightarrow yz = xz + xy$
We can take x common from the terms on the right-hand side of the equation.
$\Rightarrow yz = x(z + y)$
Rearranging the terms, we can find the value of x:
$\Rightarrow x = \dfrac{{yz}}{{y + z}}$

Therefore, the value of x is $x = \dfrac{{yz}}{{y + z}}$

Additional information:
Common errors one might make are:
Error in canceling out the terms in the equation. For example, one might forget to cancel x in both the numerator and denominator.
Error in taking common. One might forget to take the variable or number common from one of the multiple terms into consideration. This might further lead to errors in simplification.

Note:
Alternative method:
The given question is $\dfrac{1}{x} = \dfrac{1}{y} + \dfrac{1}{z}$ . In order to solve for x, we have to isolate the variable, x from y and z.
Taking LCM on the right-hand side, $\dfrac{1}{x} = \dfrac{{z + y}}{{yz}}$
Cross multiplying the terms in the expression above, $yz = x(z + y)$
Rearranging the terms, we obtain the value of x as follows, $x = \dfrac{{yz}}{{y + z}}$
Therefore, the value of x is $x = \dfrac{{yz}}{{y + z}}$