Problem 70 - A Graphical Approach to Precalculus with Limits

Question

Solve each formula for the specified variable. \(P=a+b+c\) for \(c \quad\) (Perimeter of a triangle)

Step-by-Step Solution

Answer

The formula solved for \(c\) is \(c = P - a - b\).

Step 1: Identify the Equation

The given equation represents the perimeter of a triangle: \(P = a + b + c\). Our goal is to solve for the variable \(c\).

Step 2: Isolate the Variable

To isolate \(c\), subtract \(a\) and \(b\) from both sides of the equation: \(P - a - b = c\).

Step 3: Rearrange the Equation

Rearrange the equation to clearly express \(c\) as a function of \(P\), \(a\), and \(b\): \(c = P - a - b\).

Key Concepts

Perimeter of a Triangle

The concept of the perimeter of a triangle is fundamental in geometry. The perimeter refers to the total distance around the triangle, which is calculated by summing up the lengths of all three sides. Imagine you are walking along the edges of a triangular park. The distance you cover is essentially the perimeter of the triangle.

For a triangle with sides labeled as \( a \), \( b \), and \( c \), the perimeter \( P \) can be expressed mathematically as:

• \( P = a + b + c \)

Understanding the perimeter is essential for solving problems related to boundary measurements in various real-life contexts. Whether you are dealing with land plots or designing objects, knowing how to calculate the perimeter will always be useful.

Isolate Variable

Isolating a variable is a critical step in algebra when we need to express one variable in terms of others. It is like cleaning up a room to find a specific item you need.

In our exercise, we aim to isolate the variable \( c \) from the equation \( P = a + b + c \). The process involves using operations that simplify the equation while maintaining its balance, much like keeping a seesaw level when you shift weight from one side to the other.

• Subtract \( a \) from both sides: \( P - a = b + c \)
• Then, subtract \( b \) from both sides: \( P - a - b = c \)

The result \( c = P - a - b \) shows us \( c \) on its own, allowing us to see clearly how its value depends on \( P \), \( a \), and \( b \). Mastering the skill of isolating variables unlocks the ability to solve for any unknown in similar contexts.

Rearranging Equations

Rearranging equations is like organizing your files to easily find and use specific information. This is a vital skill in both mathematics and problem-solving. By rearranging, we change the form of an equation to make it easier to work with or to better suit our needs.

When we say rearrange, we often mean modifying the order and format of the equation without altering its fundamental relationships or equality. For instance, in the equation \( P = a + b + c \), when our goal is to express \( c \), we rearrange it to:

• \( c = P - a - b \)

This version clearly shows \( c \) as a function of the other variables, making it straightforward to determine \( c \) for any given values of \( P \), \( a \), and \( b \). Learning to rearrange equations is like acquiring a toolkit that enhances your ability to solve a variety of mathematical and practical problems efficiently.