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Cross-multiplication is a powerful technique for solving equations involving fractions. It helps to eliminate the denominators, making the equation easier to solve. To cross-multiply, we multiply the numerator of one fraction by the denominator of the other fraction and set the products equal to each other. For example, in the equation \( \frac{9}{10} = \frac{1}{y} \), cross-multiplying gives us: 9 \( \times \) y = 10 \( \times \) 1. This simplifies to 9y = 10. Cross-multiplication turns a fraction equation into a linear equation, which is simpler to work with.

Isolating the variable is a key step in solving equations. It means getting the variable alone on one side of the equation. In the equation \( 9y = 10 \) from our cross-multiplication example, we want to isolate \( y \). To do this, we divide both sides by 9, which gives us: \( y = \frac{10}{9} \). By isolating \( y \), we have solved the equation and found that \( y \) must be \( \frac{10}{9} \) for the original equation to hold true. Remember, isolating the variable often involves undoing operations like addition, subtraction, multiplication, or division that are applied to the variable.

Understanding simple fractions like \( \frac{9}{10} \) and \( \frac{1}{y} \) is crucial for solving rational equations. A fraction consists of a numerator (top number) and a denominator (bottom number). In the given equation, the fractions compare two different values. To solve, we first convert the equation into a different form using cross-multiplication. Knowing that a fraction represents division is helpful. \( \frac{9}{10} \) simply means 9 divided by 10. Simplifying fractions and understanding their properties can make it easier to work with rational equations. Always remember, when solving equations with fractions, check your solutions by plugging them back into the original equation to ensure they satisfy the equation.