Cross multiplication is a technique used to solve proportions and rational equations. It involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa, to determine if the two fractions are equivalent or to solve for an unknown value.
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Cross multiplication is used to determine if two fractions are equivalent by setting the cross products (the products of the numerator of one fraction and the denominator of the other) equal to each other.
When solving rational equations, cross multiplication is used to eliminate the denominators and transform the equation into a polynomial equation that can be solved using algebraic methods.
Cross multiplication can be used to solve for an unknown value in a proportion by setting the cross products equal and solving for the unknown term.
The process of cross multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa, to determine if the two fractions are equivalent or to solve for an unknown value.
Cross multiplication is a fundamental technique in working with fractions, rational expressions, and proportions, and is essential for understanding and solving a wide range of algebraic problems.
Review Questions
Explain how cross multiplication can be used to determine if two fractions are equivalent.
To determine if two fractions, a/b and c/d, are equivalent using cross multiplication, you would multiply the numerator of the first fraction (a) by the denominator of the second fraction (d), and the numerator of the second fraction (c) by the denominator of the first fraction (b). If the resulting cross products are equal (ad = bc), then the two fractions are equivalent. This technique allows you to quickly compare the relative values of two fractions without having to convert them to a common denominator.
Describe the process of using cross multiplication to solve a rational equation.
When solving a rational equation, such as $\frac{2x + 1}{x - 3} = \frac{x + 5}{x + 2}$, cross multiplication can be used to eliminate the denominators and transform the equation into a polynomial equation that can be solved using standard algebraic methods. To do this, you would 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 fraction, setting the resulting cross products equal to each other: $(2x + 1)(x + 2) = (x + 5)(x - 3)$. This allows you to solve for the unknown variable, x, by expanding the equation and solving the resulting polynomial.
Analyze how cross multiplication can be used to solve for an unknown value in a proportion.
In a proportion, such as $\frac{a}{b} = \frac{c}{d}$, where one of the values is unknown, cross multiplication can be used to solve for the unknown term. To do this, you would 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 fraction, setting the resulting cross products equal to each other: $ad = bc$. By rearranging this equation, you can solve for the unknown value in the proportion. For example, if the proportion is $\frac{x}{6} = \frac{10}{15}$, cross multiplying and solving for x gives $x = 10$.