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When delving into complex numbers, we encounter expressions that have both real and imaginary components. Complex numbers are typically written in the form of a + bi, where a is the real part, and bi is the imaginary part, with i being the square root of -1.

In the educational exercise provided, for the complex number 2x - yi, the real part is 2x and the imaginary part is -yi. Conversely, in the complex number 14 + 12i, the real part is 14, and the imaginary part is 12i. Understanding this distinction is crucial as it forms the basis for equating and solving equations that involve complex numbers.

Solving complex equations typically involves finding the values of the unknowns that make the equation true. The process often requires setting the real parts equal to each other and the imaginary parts equal to each other, as they operate independently within a complex equation.

For example, in the exercise at hand, we are given 2x - yi = 14 + 12i. The solution involves two critical steps: isolating and equating the real components (2x and 14), which allows us to solve for x, and then doing the same for the imaginary components (-yi and 12i) to find the value of y. Once the values are found, we can confirm the solution by substituting back into the original equation to ensure both sides balance.

Coined as 'equating coefficients', this method is pivotal in confirming that each corresponding part of the complex equations is balanced. For the real parts to be equal, their coefficients must match, and the same goes for the imaginary parts.

In our given exercise, by equating the coefficients of the real part, we get 2x = 14, yielding x = 7. Next, we compare the coefficients of the imaginary part, resulting in -y = 12, thus y = -12.

This approach not only makes solving such equations more straightforward but also feeds into a broader understanding of complex number algebra. By grasping the concept of 'equating coefficients,' students can more easily navigate through complex algebraic problems and unravel solutions step by step.