Tell what can be done to both sides of the equation a d=b c...

Tell what can be done to both sides of the equation $$ a d=b c $$ to get each of the following equations. $$\text { Example: } \frac{a}{b}=\frac{c}{d}$$ $$ \begin{array}\text{Answer: Divide} \text{ by } { bd } \\ \frac{a d}{b d}=\frac{b c}{b d}, \text { and so } \frac{a}{b}=\frac{c}{d} \end{array} $$ $$\frac{d}{c}=\frac{b}{a}$$

Tell what can be done to both sides of the equation
$$
a d=b c
$$
to get each of the following equations.
$$\text { Example: } \frac{a}{b}=\frac{c}{d}$$
$$
\begin{array}\text{Answer: Divide} \text{ by } { bd } \\
\frac{a d}{b d}=\frac{b c}{b d}, \text { and so } \frac{a}{b}=\frac{c}{d}
\end{array}
$$
$$\frac{d}{c}=\frac{b}{a}$$

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Key Concepts

Manipulating Ratios and Proportions

Transforming equations into ratios or proportions, such as converting a product equation into expressions like a/b = c/d or d/c = b/a, relies on the careful use of division. This concept is essential in understanding how two relationships are equal in proportion and how to restructure equations to highlight these relationships.

Properties of Equality

This concept involves the understanding that performing the same operation on both sides of an equation preserves its equality. Whether you add, subtract, multiply, or divide by the same nonzero number (or expression), the relationship between the two sides remains unchanged, allowing for equivalent transformations.

Division as an Algebraic Operation

Division is used to simplify equations by removing common factors from both sides. When you divide both sides of an equation by a specific product of terms, you manipulate the equation into a different, yet equivalent, form. This step is crucial when solving for a variable or expressing the equation as a ratio.

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