Problem 50 - Algebra A Combined Function

Question

Solve each equation for the indicated variable. \(C=\frac{D(A+1)}{24}\) for \(A\) (Medicine: Cowling's Rule for child's dose)

Step-by-Step Solution

Answer

The solution for \(A\) is \(A = \frac{24C}{D} - 1\).

Step 1: Eliminate the Fraction

To get rid of the fraction in the equation, multiply both sides by 24. This gives us: \[24C = D(A + 1)\]

Step 2: Isolate Terms with A

Next, we want to isolate the terms containing \(A\). Divide both sides by \(D\) to achieve this:\[\frac{24C}{D} = A + 1\]

Step 3: Solve for A

Subtract 1 from both sides to solve for \(A\):\[A = \frac{24C}{D} - 1\]

Key Concepts

Equation Solving

Solving an equation involves finding the value of an unknown variable that makes the equation true. To solve equations, we need to perform operations that simplify and ultimately isolate the variable.
Basic equation solving steps typically consist of:

• performing operations like addition, subtraction, multiplication, or division on both sides of the equation,
• simplifying expressions wherever possible,
• and continuously reiterating these steps until the desired variable is isolated.

Understanding how to manipulate equations precisely is key. For example, in Cowling's Rule for child's dosage, given by the equation \(C = \frac{D(A+1)}{24}\), solving for \(A\) requires different techniques and meticulous operation on the equation. By practicing systematic equation solving, one builds the proficiency needed to tackle even more complex problems.

Isolating Variables

Isolating a variable means rewriting an equation so that a particular variable appears by itself on one side of the equation. This is often done to solve for that variable, making it easier to understand its behavior or calculate its value.
To isolate \(A\) in the equation \(C = \frac{D(A+1)}{24}\), we first made sure to eliminate other terms and factors surrounding \(A\). This included steps like:

• Clearing fractions,
• Dividing or multiplying to separate the variable from coefficients, and
• Adding or subtracting to set the variable alone.

Ultimately, these operations provided us with \(A = \frac{24C}{D} - 1\). By following these steps, we transform the equation into a straightforward statement about \(A\), making it interpretable and actionable.

Fraction Elimination

Fractions often complicate equations, making them less straightforward to work with. Eliminating fractions simplifies the process of solving equations.
To eliminate fractions, a common strategy is to multiply every term in the equation by the denominator. This is because multiplying by the denominator effectively cancels the fraction.
In the given equation, \(C = \frac{D(A+1)}{24}\), multiplying both sides by 24 clears the fraction, resulting in \(24C = D(A+1)\). By removing this fraction early in the problem-solving process, the equation becomes easier to manipulate and solve further.