Problem 46 - Elementary and Intermediate Algebra

Question

Solve each formula for the specified variable. $$ P=L+\frac{s}{f} i \text { for } s $$

Step-by-Step Solution

Answer

The formula for \( s \) is \( s = \frac{f(P - L)}{i} \).

Step 1: Isolate the Fraction

Start by isolating the fraction \( \frac{s}{f} \) on one side of the equation. Subtract \( L \) from both sides to achieve this:\[ P - L = \frac{s}{f} i \]

Step 2: Solve for \( \frac{s}{f} \)

To make \( \frac{s}{f} \) the subject, divide both sides by \( i \):\[ \frac{s}{f} = \frac{P - L}{i} \]

Step 3: Solve for \( s \)

To solve for \( s \), multiply both sides by \( f \):\[ s = f \left( \frac{P - L}{i} \right) \] or equivalently,\[ s = \frac{f(P - L)}{i} \]

Key Concepts

Solving equations

When we talk about solving equations, we're referring to finding the value or values that satisfy the equation. Think of an equation as a balance scale, where both sides need to hold the same weight. In our given problem, we want to find the value of \( s \) in terms of other variables like \( P, L, i, \) and \( f \).
Understanding this concept is like solving a mystery—everything depends on unravelling the clues (or numbers) that maintain the balance on both sides of the equation. We achieve this by simplifying the equation sequentially through various mathematical operations until the target variable is isolated.

• Start by identifying the variable you need to solve for.
• Use mathematical operations like addition, subtraction, multiplication, or division to simplify and balance the equation until you isolate that variable.

Remember, every action you perform on one side, you must also do on the other to maintain balance.

Isolating variables

Isolating a variable is like gathering all pieces of a puzzle related to one side of the board. It's necessary when you want to find out the value of a specific variable in terms of everything else.\(\)In the given equation \( P = L + \frac{s}{f} i \), our target is to express \( s \) isolated from everything else on one side of the equation.
The process involves:

• First, move all terms not containing the variable to the other side. In our equation, this means getting \( L \) on the opposite side of \( \frac{s}{f} i \), by subtracting \( L \) from both sides:

\[ P - L = \frac{s}{f} i \]

• Then, since \( s \) is also divided by \( f \) and multiplied by \( i \), these operations must also be undone. This is achieved by dividing and then multiplying by these factors:

By gradually peeling away the surrounding layers through opposite operations, you isolate \( s \). This ensures \( s \) is finally set free from its algebraic grips.

Fraction manipulation

Fraction manipulation can sometimes seem intimidating, but it's quite straightforward with the right approach. In our equation, part of solving for \( s \) involves dealing with a fraction, \( \frac{s}{f}i \). Understanding how to manipulate fractions is key.
Fraction manipulation means performing operations like addition, subtraction, multiplication, and division on fractions to simplify them. Here's what we need in our example:

• Initially, \( \frac{s}{f} \) is linked to \( i \) in the equation. Start by dividing the entire other side by \( i \) to isolate this fraction:

\[ \frac{s}{f} = \frac{P - L}{i} \]

• To further simplify and solve for \( s \), we eliminate the fraction by multiplying both sides by \( f \):

\[ s = f \left( \frac{P - L}{i} \right) \]

• This converts our fraction into a neat equation for \( s \), making any additional computations straightforward.