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When solving rational equations, cross-multiplication is a powerful technique to eliminate fractions. It involves multiplying across the equal sign by pairing the numerator of one fraction with the denominator of the other, and vice versa.
This method is especially useful when you are dealing with two fractions set to be equal. For instance, if you have \( \frac{a}{b} = \frac{c}{d} \), you can cross-multiply to get \( a \cdot d = b \cdot c \).
This step effectively removes the denominators and leaves you with a quadratic equation to solve, as seen in the original problem where \( 4(2t - 1) = (3t + 2)(2) \). Keep in mind that cross-multiplication can only be used when you have exactly one fraction on each side of the equation.

Once you've cross-multiplied and simplified the equation to the point where it's free of fractions, as in \( (4)(2t - 1) = (3t + 2)(2) \), the next step is to use the distributive property. This mathematical principle allows us to multiply a term by each term within a bracket or parentheses.
In our example, apply the distributive property to both sides: multiply \(4\) through \(2t - 1\) to get \(8t - 4\) and multiply \(2\) through \(3t + 2\) to obtain \(6t + 4\).
Essentially, it involves:

• Converting expressions such as \(a(b + c)\) into \(ab + ac\).
• Simplifying complex algebraic expressions into manageable parts.

Remember, mastering the distributive property, just like in the given problem, often forms a crucial step in transforming equations into simpler, solvable forms.

After distributing terms, the next step involves isolating the variable, which in our example is 't'. Solving the equation \(8t - 4 = 6t + 4\) requires us to get all terms involving 't' on one side.
Start by subtracting \(6t\) from both sides, simplifying it to \(2t - 4 = 4\).
Isolating the variable means making the variable the subject of the formula. To completely isolate \(t\), add or subtract terms as necessary, ensuring that the variable is by itself on one side of the equation.

• In our case, after subtracting, you add \(4\) to both sides, resulting in \(2t = 8\).
• The final step is dividing both sides by \(2\) to solve for \(t\), so \(t = 4\).

Isolating variables is crucial in solving equations as it leads directly to the solution, clearly illustrated in the final step of the problem.