Problem 64 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The solution for \(c\) is \(c = p - a - b\).

Step 1: Identify the Equation

We start with the algebraic equation that represents the perimeter of a triangle: \( p = a + b + c \).

Step 2: Isolate the Variable

To solve for \( c \), we need to isolate it on one side of the equation. We do this by subtracting \( a + b \) from both sides of the equation: \( p - a - b = c \).

Step 3: Verify the Solution

Ensure the solution is correct by checking that \( c = p - a - b \) maintains balance in the equation. By doing this, we understand that we correctly rearranged the formula to solve for \( c \).

Key Concepts

Perimeter of a Triangle

The perimeter of a triangle is determined by summing the lengths of its three sides. This is a basic concept in geometry and can be represented by the equation:

• \( p = a + b + c \)

Where:

• \( p \) stands for the perimeter of the triangle,
• \( a \), \( b \), and \( c \) are the lengths of its sides.

Understanding this formula is crucial because it provides a simple way to calculate the complete distance around the triangle.
Perimeters are widely used not just in mathematical problems, but also in real-life scenarios, such as determining the amount of material needed to create a border around an object.

Isolating Variables

Isolating a variable means rearranging an equation so that the variable is alone on one side. This allows you to solve for that specific variable. For example, if you're given the equation representing the perimeter of a triangle, \( p = a + b + c \), and you need to solve for \( c \), you have to perform some operations:

• Subtract \( a \) and \( b \) from both sides to isolate \( c \).

After these operations, you're left with:

• \( c = p - a - b \)

This process is all about balancing the equation, which is a foundational skill in algebra. Think of an equation as a balanced scale; any action you take on one side must be mirrored on the other to maintain balance. This principle guides the manipulation of equations to find the desired variable.

Equation Rearrangement

Equation rearrangement involves changing the structure of an equation to better understand or solve for a specific part of it. When rearranging the perimeter formula \( p = a + b + c \) to solve for \( c \), you adjust the equation to

• \( c = p - a - b \)

Essentially, you are performing the inverse operations to isolate \( c \). Rearrangement is key in problem-solving as it allows you to transform a complicated equation into a simpler one, making it easier to interpret and work with.
Rearranging equations is a powerful technique that applies not just to geometry but across all branches of mathematics and science. Being comfortable with rearranging formulas can be very helpful in understanding complex systems.