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Cross multiplication is a fundamental technique used to solve proportions, which are equations that set two fractions equal to each other. For instance, consider the proportion \(\frac{2}{5} = \frac{3x}{7}\). When you cross multiply, you multiply the numerator of one fraction by the denominator of the other and vice versa. Here's how to carry it out step by step:

• Multiply the numerator of the first fraction (2) by the denominator of the second fraction (7).
• Simultaneously, multiply the numerator of the second fraction (3x) by the denominator of the first fraction (5).

Once both products are calculated, you set them equal to each other, as shown: \(2 \times 7 = 3x \times 5\). This technique effectively clears the fractions, allowing for easier manipulation of the equation in latter steps.

To find the value of an unknown variable, we need to 'isolate' it on one side of the equation. Isolating a variable means performing operations that will leave the variable by itself on one side of the equal sign, making it the subject of the formula.

For the equation \(14 = 15x\), our goal is to isolate \(x\). This is done by performing the inverse operation of whatever is applied to the variable. Since \(x\) is multiplied by 15, we divide both sides of the equation by 15, resulting in \(x = \frac{14}{15}\).

It's crucial to remember that whatever you do to one side of the equation, you must do to the other to maintain equality. This preserves the balance of the equation, which is essential for finding the correct value of the variable.

Fractional equations are equations that include fractions and often involve variables either in the numerator, the denominator, or both. Solving fractional equations generally involves finding a common denominator, cross multiplying to eliminate the fractions, and then solving the resulting equation.

In our example, \(\frac{2}{5} = \frac{3x}{7}\), we use cross multiplication as the preferred method since it removes the fractions quickly. After cross multiplying and simplifying, we obtain an equation without fractions, allowing us to use standard algebraic techniques to solve for \(x\).

Working with fractional equations requires careful arithmetic and algebraic manipulation. It's important to be comfortable with fractions, operations on them, and knowing methods like cross multiplication, which strengthen your problem-solving toolkit when tackling these types of equations.