Question

How do you solve for $ z $ in $ {y^2} + 3yz - 8z - 4x = 0 $ ?

Answer 1

Hint: In the given equation we can see that we have three variables given in one equation. We cannot find the value of the variables using only one equation. Solving for $ z $ means finding the expression of $ z $ in the form of all other variables after simplification of the equation.

Complete step by step solution:
We have been given an equation $ {y^2} + 3yz - 8z - 4x = 0 $ containing three variables in a single equation. We cannot find the value of $ z $ . Solving the equation means writing $ z $ in terms of other variables in the equation.
First we will try to keep all the terms containing $ z $ on the left side of the equation and transfer all other terms to the right side.
We will subtract $ {y^2} $ from both sides of the equation.
 $ \Rightarrow 3yz - 8z - 4x = - {y^2} $
Now we add $ 4 $ to both the sides.
 $ \Rightarrow 3yz - 8z = - {y^2} + 4x $
On the left side we are left with all the terms containing $ z $ . We can take $ z $ common from both the terms.
 $ \Rightarrow z\left( {3y - 8} \right) = - {y^2} + 4x $
We divide each side by $ \left( {3y - 8} \right) $ (assuming that $ y $ should not be equal to $ \dfrac{8}{3} $ ).
 $ \Rightarrow z = \dfrac{{\left( {4x - {y^2}} \right)}}{{\left( {3y - 8} \right)}} $
Thus, we get the final expression where $ z $ is represented in the form of other variables.
Hence, $ z = \dfrac{{\left( {4x - {y^2}} \right)}}{{\left( {3y - 8} \right)}} $ is the solution for $ z $ .
So, the correct answer is “ $ z = \dfrac{{\left( {4x - {y^2}} \right)}}{{\left( {3y - 8} \right)}} $ ”.

Note: We can find the value of $ z $ by knowing the values of $ x $ and $ y $ . While transferring the terms to the other side of an equation the sign of the term inverses, i.e. ‘+’ becomes ‘-‘ and vice-versa. We mentioned that $ y $ should not be equal to $ \dfrac{8}{3} $ because otherwise $ z $ will become undefined. While dividing a term by any term we have to note that the denominator should not be equal to zero.