Problem 26 - Algebra Form and Function

Question

In Exercises 26-37, solve for the indicated variable. Assume all constants are non-zero. $$ A=l \cdot w, \text { for } w $$

Step-by-Step Solution

Answer

Answer: The formula to find the width (w) of a rectangle, given its area (A) and length (l), is w = A / l.

Step 1: Understand the given equation

The given equation is A = lw. This represents the area (A) of a rectangle as the product of its length (l) and width (w).

Step 2: Isolate width (w)

To solve for w, we need to isolate it on one side of the equation. In this case, we should divide both sides of the equation by l to get w by itself. This will look like: $$ w = \frac{A}{l} $$

Step 3: Write the final formula

After isolating w, we get the formula for width in terms of length and area: $$ w = \frac{A}{l} $$ This is the final formula to solve for the width (w) of a rectangle, given its area (A) and length (l).

Key Concepts

Algebra

When we talk about algebra, we're diving into a mathematical language that helps us describe relationships using symbols and letters. It's like creating equations or formulas to represent real-world scenarios or problems. The very first thing to understand about algebra is that it's all about finding the unknown. We use letters, like \( w \),\( l \), or \( A \), to represent unknowns or variables and numbers to express known values.
Algebra allows you to manipulate these symbols to find what that unknown number is. It can involve operations like addition, subtraction, multiplication, and division. For example, in the equation for the area of a rectangle, \( A = l \cdot w \), we see how algebra uses symbols to generalize a specific mathematical relationship.

• The equation expresses area as a product of two variables (length and width).
• Solving involves working out what one variable equals when the others are known.

So, algebra is not just about numbers; it's about figuring out how numbers relate to each other.

Rectangular Area

The area of a rectangle is an important concept because rectangles are such common shapes, both practically and geometrically. The area tells us how much space is inside the rectangle. Imagine you have a piece of fabric in the shape of a rectangle, and you want to know how much it can cover—this is where knowing the area is crucial.
To calculate the area of a rectangle, we use the formula \( A = l \cdot w \), where \( l \) stands for length and \( w \) stands for width. This equation shows us that:

• The area depends on both the length and the width.
• Doubling either the length or width will double the area.

The concept of area extends beyond understanding the size of surfaces; it helps in solving problems related to space usage, carpeting rooms, building layouts, and more. Essentially, it lets us quantify space in a meaningful way.

Variable Isolation

Variable isolation is a fundamental technique in algebra that lets us solve equations more easily by getting the unknown variable by itself on one side. Imagine you are isolating the width, \( w \), from the area equation \( A = l \cdot w \). To do this, you must get \( w \) alone, which involves moving the other terms to the opposite side of the equation.
In our example, you achieve this by dividing both sides of the equation by \( l \) (the length). This gives:

• The formula \( w = \frac{A}{l} \), representing the width in terms of area and length.
• This means you can find the width if you know the area and length, by simply doing a division.

Variable isolation isn't limited to rectangles or basic formulas. It's a strategy used in all sorts of equations, where the aim is to make calculations simpler by focusing on one variable at a time. This makes solving complex problems much more manageable.