Question

List some formulas that occur in applications. Solve each formula for the indicated variable. $$ \text { Mathematics } S=\frac{a}{1-r} \text { for } r $$

Short answer

Formula solved for \( r \): \( r = \frac{S - a}{S} \)

Step-by-step solution

Write Down the Given Formula

Start with the given formula in mathematics: \[ S = \frac{a}{1-r} \]

Isolate the Denominator

To isolate the variable \( r \), first manipulate the formula to eliminate the denominator. Multiply both sides of the equation by \(1 - r\): \[ S(1 - r) = a \]

Expand and Rearrange the Equation

Distribute \( S \) on the left side of the equation: \[ S - Sr = a \]

Solve for \( r \)

Move all terms involving \( r \) to one side of the equation and constant terms to the other side. Subtract \( S \) from both sides: \[ -Sr = a - S \] Next, divide by \( -S \) to isolate \( r \): \[ r = \frac{S - a}{S} \]

Key concepts

algebraic manipulation

Algebraic manipulation is a fundamental skill in mathematics.
This involves rearranging and simplifying expressions.
Algebraic manipulation helps solve equations or make them easier to understand.

In the given exercise, we start by working with the formula: \[ S = \frac{a}{1-r} \]

The goal is to rearrange this formula to solve for the variable \( r \).
The first step involves multiplying both sides by \( (1-r) \) to eliminate the fraction.
This action is a classic example of algebraic manipulation.

When we multiply out the expression, new terms are introduced:
\[ S(1-r) = a \]

From here, distributing \( S \) on the left gives us:
\[ S - Sr = a \]
This reorganization is another example of algebraic manipulation in action.

isolating variables

Isolating variables means making one variable the subject of an equation.
In the exercise, we need to solve for \( r \).
To do this, we aim to have \( r \) alone on one side of the equation.

Starting from the manipulated equation:
\[ S - Sr = a \]
We need to move all terms involving \( r \) to one side.
Subtract \( S \) from both sides:
\[ S - S - Sr = a - S \]
Which simplifies to:
\[ -Sr = a - S \]
Next, we divide by \( -S \) to isolate \( r \):
\[ r = \frac{S - a}{S} \]
Now, the variable \( r \) is isolated and we have our solution.
Properly isolating variables is key to simplifying and solving equations.

fractional equations

Fractional equations contain fractions and often require special steps to solve.
These equations can be challenging because we need to clear the fractions first.

For example, our exercise starts with the equation:
\[ S = \frac{a}{1-r} \]
Here, \( a \) is divided by \( 1-r \).
The first step to solving this is to eliminate the fraction by multiplying both sides by the denominator:
\[ S(1-r) = a \]
This clears the denominator and simplifies the equation dealing directly with the variables:
\[ S - Sr = a \]
Notice that after eliminating the fraction, we can use standard algebraic manipulation techniques to isolate the variable.
Understanding how to manage and manipulate fractional equations opens up many mathematical problems to simpler solutions.