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To solve a proportion equation like \(\frac{12}{4} = \frac{x}{3}\), cross-multiplication can be a very handy tool. Cross-multiplication is like creating two new equations by switching and multiplying the denominators with the numerators from the other fraction.
This step helps in removing the fractions and getting a simpler equation to work with.
Here's how it works:

• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Do the same for the numerator of the second fraction and the denominator of the first fraction.

So, for our equation: \[ 12 \times 3 = 4 \times x \]After cross-multiplying, the fractions are gone, and we have a simple linear equation to solve. This step is essential for making the equation easier to handle and takes us closer to the solution.

Once we have cross-multiplied and formed our new equation, the next step is to simplify it. Starting from: \[ 12 \times 3 = 4 \times x \] Carry out the multiplication: On the left side, \(\text{12} \times \text{3} \text{ simplifies to} \ 36\), and on the right side, \(\text{4} \times \text{x} \text{ remains as} \ 4x\).
This simplifies our equation to: \[ 36 = 4x \]This step is straightforward but crucial because it makes the equation clearer and ready for the final step. Always remember to perform each operation carefully to avoid mistakes. Simplifying may also involve combining like terms or reducing fractions in more complex equations.

In the final step, our goal is to isolate the variable we're solving for to find its value. Using the simplified equation: \[ 36 = 4x \] We need to isolate \(x \). This involves performing operations that leave \( x \) alone on one side of the equation. For our equation, we achieve this by dividing both sides by 4:
\[ x = \frac{36}{4} \] When we divide both sides, we get: \[ x = 9 \] And there, we have found our solution! Isolating variables can require different operations like addition, subtraction, multiplication, or division depending on the equation. Always ensure you perform the same operation on both sides to maintain equality.
This step brings us to the final answer, making sure we have correctly solved the proportion.