Problem 10 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

For \(b\): \(b = P - a - c\) and for \(c\): \(c = P - a - b\).

Step 1: Understand the Problem

Given the formula for the perimeter of a triangle: \[P = a + b + c\]We need to solve for two different variables: (a) for \(b\) and (b) for \(c\).

Step 2: Solving for b

To isolate \(b\), subtract \(a\) and \(c\) from both sides of the equation:\[P = a + b + c\]Subtract \(a\) from both sides:\[P - a = b + c\]Subtract \(c\) from both sides:\[P - a - c = b\]So, the solution for \(b\) is:\[b = P - a - c\]

Step 3: Solving for c

To isolate \(c\), subtract \(a\) and \(b\) from both sides of the equation:\[P = a + b + c\]Subtract \(a\) from both sides:\[P - a = b + c\]Subtract \(b\) from both sides:\[P - a - b = c\]So, the solution for \(c\) is:\[c = P - a - b\]

Key Concepts

Perimeter of a Triangle

Understanding the perimeter of a triangle is a fundamental concept in geometry. The perimeter is the total distance around the triangle, calculated by summing the lengths of all three sides.
For a triangle with sides labeled as a, b, and c, this can be expressed as:
\[ P = a + b + c \] This formula is essential because it allows you to find the length of any side if you know the perimeter and the lengths of the other sides.
For example, to find the length of side b when the perimeter P and the lengths of sides a and c are known, you would rearrange the formula to solve for b.

Isolating Variables

In algebra, isolating variables is a crucial technique used to solve equations. It involves getting the variable you are solving for on one side of the equation by itself. Here's a look at how this is done in the context of the perimeter of a triangle.
Let's say we want to isolate b in our perimeter formula: \[ P = a + b + c \] To isolate b, we need to move the other terms (a and c) to the opposite side of the equation. We do this by performing the same operation on both sides of the equation.
Step-by-step, it looks like this: \[ P - a = b + c \] (we subtracted a from both sides) \[ P - a - c = b \] (we subtracted c from both sides)
This process leaves b by itself: \[ b = P - a - c \]
Now, b is isolated, and we have a formula to calculate its value given the perimeter and the other two sides.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging equations to solve for a particular variable. This often involves basic arithmetic operations like addition, subtraction, multiplication, and division. Let's explore this concept further using our perimeter example.
Suppose we need to solve for c in the perimeter formula: \[ P = a + b + c \] Just like before, our goal is to rearrange the equation so that c is isolated.
We begin by subtracting a from both sides: \[ P - a = b + c \] Next, we subtract b from both sides: \[ P - a - b = c \] And we've isolated c: \[ c = P - a - b \]
Through systematic algebraic manipulation, we've derived a formula that allows us to calculate c if the perimeter P and the lengths of a and b are known.
This method can be applied to various algebraic equations, making it an essential skill in solving mathematical problems.