Cross multiplication is a technique used to solve for an unknown variable in a proportion or rational equation. It involves multiplying the numerator of one fraction by the denominator of the other fraction, and then setting the two resulting products equal to each other.
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Cross multiplication is a useful tool for solving for an unknown variable in a proportion or rational equation.
The cross multiplication technique involves setting the product of the numerator of one fraction and the denominator of the other fraction equal to the product of the numerator of the other fraction and the denominator of the first fraction.
Cross multiplication can be used to simplify rational expressions by finding common factors and canceling them out.
Proportions and rational equations often arise in real-world applications, such as in the context of rates, unit conversions, and probability.
Mastering cross multiplication is essential for solving a variety of algebraic problems involving fractions and proportions.
Review Questions
Explain the process of cross multiplication and how it can be used to solve for an unknown variable in a proportion.
Cross multiplication is a technique used to solve for an unknown variable in a proportion, which is a statement of equality between two ratios. The process involves multiplying the numerator of one fraction by the denominator of the other fraction, and then setting the two resulting products equal to each other. This allows you to isolate the unknown variable and solve for its value. For example, if you have the proportion $\frac{a}{b} = \frac{c}{d}$, you can cross multiply to get $ad = bc$, and then solve for the unknown variable.
Describe how cross multiplication can be used to simplify rational expressions.
Cross multiplication can be used to simplify rational expressions by finding common factors between the numerator and denominator and canceling them out. This is done by setting the cross products of the fractions equal to each other, which allows you to isolate and eliminate the common factors. For instance, if you have the rational expression $\frac{3x^2 - 9x}{x^2 - 3x}$, you can cross multiply to get $3x^2 - 9x = x^2 - 3x$, and then cancel out the common factor of $x - 3$ in the numerator and denominator to simplify the expression.
Analyze how the cross multiplication technique can be applied to solve real-world problems involving proportions and rational equations.
Cross multiplication is a powerful tool for solving a variety of real-world problems that involve proportions and rational equations. For example, in the context of unit conversions, you can use cross multiplication to set up a proportion and solve for an unknown unit of measurement. In the realm of probability, cross multiplication can be used to solve problems involving the relationship between the probability of an event and its complement. Additionally, cross multiplication is essential for solving problems related to rates, such as calculating the speed of an object or the time it takes to complete a task. By understanding the underlying principles of cross multiplication and how to apply it in different contexts, you can effectively solve a wide range of algebraic problems encountered in college algebra and beyond.
A proportion is a statement of equality between two ratios, where the relationship between the numerator and denominator is the same in both fractions.
Rational Equation: A rational equation is an equation that contains one or more rational expressions, which are fractions with variables in the numerator, denominator, or both.
Equivalent Fractions: Equivalent fractions are fractions that have the same value, even though the numerator and denominator may be different.