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Cross multiplication is a technique used to solve proportions, which are statements that two ratios are equal to each other. It involves multiplying across the equal sign, so you multiply the numerator of one ratio by the denominator of the other ratio and vice versa. For example, if you have a proportion like \[\frac{a}{b} = \frac{c}{d},\] you would cross multiply to get \[ad = bc.\] This step effectively removes the fractions and gives you a simpler equation to work with.

When we look at the exercise involving the proportion \[\frac{z}{35} = \frac{5}{8},\] cross multiplication is applied to find the value of the variable 'z'. Multiplying 8 by 'z' and 35 by 5, we arrive at \[8z = 175.\] From here, we can more easily solve for 'z', moving us closer to the solution.

Ratios are a way to compare two quantities by showing the relative size of one quantity to another. They are expressed in the form \[a : b \text{ or } \frac{a}{b},\] where 'a' is considered the first term and 'b' is the second term of the ratio. Ratios can be used to represent real-world relationships such as speed (distance per time), density (mass per volume), or even in recipes (ingredients quantity). It is crucial when working with ratios to maintain the consistency of units and to understand that ratios are a form of fraction.

In our exercise, the ratio \[\frac{z}{35}\] indicates the relationship between the variable 'z' and the number 35, and similarly, \[\frac{5}{8}\] illustrates the relationship between the numbers 5 and 8. Recognizing that proportions are equations that assert the equivalence of two ratios is key to finding the values of unknown variables.

To isolate a variable means to manipulate an equation in such a way that the variable of interest is alone on one side of the equation. This is often the ultimate goal when solving for an unknown in algebra. It involves performing the same mathematical operations on both sides of the equation in order to simplify the equation down to the form \[variable = \text{ numerical value}.\] These operations can include addition, subtraction, multiplication, division, and the application of inverse operations.

Following the cross multiplication step from our exercise, we had \[8z = 175.\] To isolate 'z', we divide both sides of the equation by 8 to get \[z = \frac{175}{8}.\] This form of the equation clearly shows the value of 'z' and completes the goal of solving the proportion. Such clear and linear steps are critical for understanding algebraic processes and arriving at correct solutions.