Problem 82 - Algebra and Trigonometry

Question

\(79-92\) Solve the equation for the indicated variable. $$ P=2 l+2 w ; \quad \text { for } w $$

Step-by-Step Solution

Answer

The solution for \(w\) is \(w = \frac{P - 2l}{2}\).

Step 1: Isolate Terms with the Variable

Start by isolating the terms that include the variable \(w\) on one side of the equation. The given equation is \(P = 2l + 2w\). We want to solve for \(w\). Subtract \(2l\) from both sides to get: \[P - 2l = 2w\]

Step 2: Solve for the Variable

Now that we have \(2w\) on one side of the equation, divide both sides by 2 to isolate \(w\). This gives us:\[w = \frac{P - 2l}{2}\]

Key Concepts

Perimeter Formula

Understanding the perimeter formula is essential in geometry. Perimeter refers to the total distance around a two-dimensional shape. The basic equation for the perimeter of a rectangle involves its length and width, and it's given by:\[ P = 2l + 2w \]Here, \(P\) represents the perimeter, \(l\) is the length, and \(w\) is the width.

• Each side of the rectangle is accounted for twice, once for each pair of opposite sides.
• This formula helps quickly determine the total length of the boundary of a rectangle.

It's a straightforward equation because it simply adds together the lengths of each side. This formula can be adapted for various applications, such as finding one missing side when the total perimeter is known, which leads us into solving equations.

Isolating Variables

When solving equations, isolating the variable of interest is a crucial step. This means rearranging the equation so that the desired variable is alone on one side. A clear methodical way to do this ensures you avoid errors:

• Identify the target variable you need to solve for—in this case, \(w\).
• Use basic arithmetic operations to move other terms to the opposite side of the equation.
• Make sure to perform the same operation on both sides to maintain equality.

In the exercise, to isolate \(w\), we subtracted \(2l\) from both sides of the equation \(P = 2l + 2w\) resulting in \(P - 2l = 2w\).Next, dividing everything by 2 simplified the expression further to \(w = \frac{P - 2l}{2}\). This ensures that \(w\) is completely isolated, providing an expression you can use to find the width when you have a perimeter and length.

Linear Equations

Linear equations are equations of the first degree, meaning each term is either a constant or the product of a constant and a single variable. These equations graph as straight lines, hence the name 'linear'.The form of a basic linear equation is:\[ Ax + B = C \]Here, \(A\), \(B\), and \(C\) are constants, and \(x\) is the variable. The goal is often to solve for \(x\) by rearranging and simplifying the equation using algebraic operations.

• They are solvable using simple methods like addition, subtraction, multiplication, or division.
• An important feature is the balance of the equation; operations done to one side must be done to the other.

In our exercise with the perimeter equation, \(P = 2l + 2w\), the equation was linear because it contains variables \(l\) and \(w\) raised only to the first power. By isolating \(w\) and simplifying the equation, we demonstrated how to solve a linear equation to make it easier to use for specific calculations, like determining an unknown side length.