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Solving proportions often involves a helpful technique called cross multiplication. This method is particularly useful when you have two fractions set equal to each other, as shown in the exercise \(\frac{8}{x+2} = \frac{3}{x-1}\). To cross multiply, you multiply the numerator of one fraction by the denominator of the other fraction. Here, you'll multiply \(8\) by \((x-1)\) and \(3\) by \((x+2)\). This gives the equation:

• \(8(x-1) = 3(x+2)\)

This technique eliminates the fractions and turns your proportion into a straightforward equation that can be solved more easily.
Using cross multiplication is a reliable way to simplify and solve proportions since it allows you to extract vital elements of the equation without fractions getting in the way.

After finding potential solutions, it's crucial to verify whether they are valid. Sometimes, while solving proportions, you'll encounter what are called extraneous solutions. These solutions result from the algebraic manipulation steps you've taken but don't actually satisfy the original proportion. Although they appear legitimate mathematically, they don't make sense when plugged back into the context of the problem.
Revisiting the solution, once we obtained \(x = \frac{14}{5}\), it was necessary to substitute it back into the original equation:

• Check if \(\frac{8}{\frac{14}{5}+2} = \frac{3}{\frac{14}{5}-1}\).

By doing this, you ensure that both sides of the proportion are equal and that your solution isn't extraneous.
Checking for extraneous solutions helps confirm that you've solved the proportion correctly and ensures the solution is practical and applicable to the problem.

Isolation of variables is a core step in solving equations. It involves manipulating the equation to have the variable you are solving for on one side by itself. Let's break down how this was done in the exercise.
After you've simplified both sides of the equation to \(8x - 8 = 3x + 6\), you need to isolate \(x\) to determine its value.

• First, subtract \(3x\) from both sides: \(5x - 8 = 6\).
• Next, add \(8\) to both sides to further isolate \(5x\): \(5x = 14\).
• Finally, divide both sides by \(5\) to solve for \(x\): \(x = \frac{14}{5}\).

Isolating variables requires systematic steps and careful calculations to ensure the equation transforms smoothly without errors. By practicing this, you can solve for the unknown effectively and gain confidence in handling different types of algebraic equations.