91Ó°ÊÓ

In order to solve rational equations, cross-multiplication is a common and effective technique. The essence of cross-multiplication is to eliminate the denominators by multiplying the numerator of each fraction by the denominator of the other. For instance, given the equation \(\frac{z}{2} = \frac{z}{y}\), cross-multiplication transforms it into \(z \cdot y = z \cdot 2\). This step simplifies the equation into a format easier to manipulate, free from fractions and thus more straightforward for solving.

Let's break it down further:

• Identify the numerators and denominators on both sides of the equation.
• Multiply the numerator of the left-hand fraction by the denominator of the right-hand fraction.
• Do the inverse: multiply the numerator of the right-hand fraction by the denominator of the left-hand fraction.

In this way, the two fractions are equated in a whole number format. It is key to ensure each side of the equation gets multiplied so the balance is maintained.

Simplifying equations makes them more manageable and easier to solve. In our given exercise, we first observed that the same expression \(y + 5\) appeared on both sides of the equation. By setting \(y + 5\) to a simpler variable, \(z\), the equation \(\frac{y+5}{2}=\frac{y+5}{y}\) was transformed into \(\frac{z}{2} = \frac{z}{y}\).

Equation simplification often involves:

• Identifying common factors or terms in various parts of the equation.
• Using substitution to replace complex terms with simpler variables.
• Calculating or canceling terms to reduce the equation to its simplest form.

This process helps highlight the core elements of the problem and reduces the load of the equation, making further steps less cumbersome and more insightful.

One of the final and critical steps in solving for a variable is isolating it on one side of the equation. After simplifying and cross-multiplying, our goal transforms into isolating \y\. Post cross-multiplication, the equation \(yz = 2z\) needs isolation of \y\.

Here is how to achieve that:

• Assure that the variable you want to isolate is not multiplied by zero to avoid undefined forms. In our solution, we assumed \z eq 0\.
• Divide both sides of the equation by the term accompanying the variable. Hence, divide by \z\ to isolate \y\.

Our division simplifies \(yz = 2z\) to \y = 2\. An isolated variable yields the solution straightforwardly. Finally, always verify by substituting back into the original equation to ensure correctness and to confirm that the solution satisfies the given conditions.