Problem 57 - Elementary and Intermediate Algebra

Question

Solve each formula for the specified variable. $$ \frac{a}{b}=\frac{c}{d} \text { for } d $$

Step-by-Step Solution

Answer

\( d = \frac{b \, c}{a} \)

Step 1: Cross-Multiply

To solve for \( d \), cross-multiply the fractions. This will give us the equation \( a \times d = b \times c \).

Step 2: Isolate \( d \)

To isolate \( d \), divide both sides of the equation by \( a \). This results in \( d = \frac{b \, c}{a} \).

Key Concepts

Cross-Multiplication

Cross-multiplication is a powerful technique when dealing with equations that contain two fractions set equal to each other, often known as fractional equations. In this process, you multiply the numerator of one fraction by the denominator of the other fraction, effectively eliminating the fractions.

• For the equation \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication involves multiplying \( a \) by \( d \) and \( b \) by \( c \).
• This results in the equation \( a \times d = b \times c \).

This method simplifies the equation by getting rid of the fractions, making it easier to solve for the desired variable. It is especially helpful because it transforms the equation into a format that allows straightforward algebraic manipulation. Cross-multiplication applies only when dealing with equal fractions and should be used with care to ensure accuracy. This step lays the groundwork for further solving the equation.

Solving for a Variable

Solving for a variable means isolating that variable on one side of the equation. When an equation is in a more complex form, like after cross-multiplication, the aim becomes clearing everything else away from the variable we need to solve for.
In our equation \( a \times d = b \times c \), where the goal is to solve for \( d \), we can use inverse operations. Here, the operation currently performed on \( d \) is multiplication by \( a \).

• To isolate \( d \), perform the inverse operation of what is currently happening. Since \( d \) is multiplied by \( a \), divide both sides of the equation by \( a \) to remove \( a \) from the side containing \( d \).
• By doing this, we obtain the equation \( d = \frac{b \, c}{a} \).

This process ensures the variable \( d \) is by itself on one side of the equation, allowing us to see directly what \( d \) equals in terms of the other variables, which is critical for clear and correct problem-solving in algebra.

Fractional Equations

Fractional equations involve fractions where the variable is in the numerator, denominator, or within the fraction itself. These equations can be intimidating at first glance because the presence of variables in fractions adds a layer of complexity.
A fractional equation appears like \( \frac{a}{b} = \frac{c}{d} \), where variables appear in fractional form. The main challenge is to find a method to simplify and clear the fractions for solving.

• Using strategies like cross-multiplication helps to eliminate the fractions, allowing us to manipulate the equation more easily.
• You can solve such equations using techniques like clearing fractions or multiplying through by a common denominator.

Once fractions are removed, solving the equation becomes more manageable, and algebraic techniques to isolate variables can be directly applied. Understanding how to approach fractional equations with logical steps enables you to tackle these problems with confidence, ensuring accurate solutions.