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Cross multiplication is a shortcut method used primarily for solving equations involving two fractions set equal to each other, which is often the case with algebraic fractions. This technique essentially allows us to eliminate the fractions by multiplying crosswise.
To employ this method, you take the numerator of the first fraction and multiply it by the denominator of the second fraction, and do the reverse for the other numerator and denominator. If we consider our example, \[\frac{5-x}{17} = \frac{13}{34},\] we cross multiply to get \[(5 - x) \times 34 = 17 \times 13.\] This step creates a simpler equation without any fractions, paving the way for further simplification and resolution.

Isolating the variable is a fundamental principle in algebra that involves manipulating the equation so the variable we’re solving for is alone on one side of the equation. This is typically done by performing operations that cancel the other terms present on the same side as the variable.
In our sample equation, \[34 \times 5 - 34x = 221,\] we start by distributing the multiplication over subtraction which simplifies to \[170 - 34x = 221.\] Next, we need to isolate 'x'. This is achieved by moving all terms without 'x' to the other side, leading to \[-34x = 51.\] Now 'x' stands alone on one side of the equation, clearly showing that it is isolated and ready for the final solving step.

The process of simplifying equations involves several techniques to manipulate and reduce equations to their simplest form. It makes the problem easier to understand and often easier to solve.
Starting with \[170 - 34x = 221,\] we execute subtraction across the equals sign \[221 - 170\] to condense this equation down to \[-34x = 51.\] This simpler form is almost ready to solve, but we first need to isolate 'x', as mentioned earlier. Once isolated, we simplify further by dividing both sides by the coefficient of 'x', which is \[-34,\] to find the value of 'x'. The operation yields \[x = -1.5,\] which is the simplest expression of the solution for 'x'. Remember, a well-simplified equation has the variable on one side and a numerical answer on the other, without any unnecessary terms or factors.