91Ó°ÊÓ

The first step in solving any equation for a particular variable is isolating that variable. In this case, we want to solve for the variable \( z \). To start, look at the equation and identify the term containing \( z \).

Here, you want to isolate \( \frac{3}{z} \) on one side. To do this, you need to get rid of the other terms from that side. In our problem:
\( 9 x+\frac{3}{z}=\frac{5}{y} \), subtract \( 9x \) from both sides: \[ \frac{3}{z} = \frac{5}{y} - 9x \].

This operation ensures that the only term left on the left side is \( \frac{3}{z} \), letting you focus solely on \( z \) for the next steps. This is crucial for solving equations.

To isolate \( z \) further, we need to understand the concept of a reciprocal. The reciprocal of a number is simply the number flipped upside down. For instance, the reciprocal of \( a \) (expressed as \( \frac{1}{a} \)) is \( \frac{1}{a} \).

In our current equation \( \frac{3}{z} = \frac{5}{y} - 9x \), z is in the denominator (bottom part of the fraction). To get \( z \) out of the denominator, we take the reciprocal of the entire equation: \[ \frac{z}{3} = \frac{1}{\frac{5}{y} - 9x} \].

Reciprocal essentially flips fractions and this becomes incredibly useful for manipulating equations, especially when you need to free a variable stuck in the denominator.

Algebraic manipulation involves rearranging and simplifying equations to isolate or solve for a variable. This can include adding, subtracting, multiplying, or dividing both sides of the equation by the same quantity.

In our exercise, after taking the reciprocal of both sides: \[ \frac{z}{3} = \frac{1}{\frac{5}{y} - 9x} \],
we need to further simplify to get \( z \) alone. Since \( z \) is divided by 3, multiplying both sides by 3 isolates \( z \):
\( z = 3 \times \frac{1}{\frac{5}{y} - 9x} \).

Through these algebraic manipulations, we can solve for a specific variable in a complex equation by breaking it down into simpler, more manageable steps.

Finally, all our efforts come to fruition as we solve for \( z \). We started with a complex equation and, through isolating variables, applying the concept of reciprocals, and effective algebraic manipulation, we reached a more straightforward form.

From \( z = 3 \times \frac{1}{\frac{5}{y} - 9x} \), we find: \[ z = \frac{3}{\frac{5}{y} - 9x} \].

And there you go; we have now isolated and solved for \( z \). Remember, these steps— isolating the variable, understanding reciprocals, algebraic manipulation, and solving for the desired variable — are essential tools for tackling various algebraic expressions. With practice, these concepts will become second nature, making complex problems much simpler to handle.