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Cross multiplication is a useful method for solving proportions. A proportion is an equation that expresses two ratios or fractions as being equal to one another. In this case, the given proportion is \( \frac{x}{10} = \frac{5}{9} \). To use cross multiplication, multiply the numerator of the first fraction by the denominator of the second fraction, and the numerator of the second fraction by the denominator of the first.

For our problem, follow these steps:

• Multiply \( x \) (the first numerator) by 9 (the second denominator).
• Multiply 5 (the second numerator) by 10 (the first denominator).

This leads to the equation \( 9x = 50 \). Cross multiplication helps eliminate the fractions, making it easier to work with whole numbers or expressions before moving on to isolating the variable. Remembering this simple multiplication across the equal sign is key to solving proportions efficiently.

Isolating variables is an essential step in solving equations. The goal is to get the unknown variable, often represented by \(x\), by itself on one side of the equation. After cross multiplying in our proportion, we obtained the equation \( 9x = 50 \).

To isolate \( x \), you need to perform operations that "undo" the multiplication by 9. Follow these steps:

• Divide each side of the equation by 9.
• This operation simplifies \( 9x = 50 \) to \( x = \frac{50}{9} \).

When you divide both sides by the same number, you maintain the equality while reducing the coefficient of the variable to 1. This step is crucial as it provides the solution and ensures that the variable is alone on one side of the equation.

Rational equations consist of ratios or fractions and can be solved using similar principles as basic algebra. They typically involve finding a common denominator or using cross multiplication to eliminate fractions.

For the equation \( \frac{x}{10} = \frac{5}{9} \), consider the cross multiplication method that aligns with rational equations involving fractions:

• Cross multiplication helps to transform the equation into a linear format without fractions.
• This method is especially beneficial when dealing with variables in the numerator.

The simplified rational equation allows for easier manipulation of terms and straightforward solving steps, such as isolating the variable, as we saw with \( x = \frac{50}{9} \). Understanding rational equations equips you to deal with more complex problems involving various fractional forms and expressions.