Problem 13 - Algebra 1: Concepts and Skills

Question

Solve the formula for the indicated variable. Show all your steps. Then evaluate the new formula by substituting the given values. Area of a rectangle: \(A=\ell w\) Solve for \(w\) Find the value of \(w\) when \(A=36\) and \(\ell=9\)

Step-by-Step Solution

Answer

The width (w) of the rectangle is 4 units.

Step 1: Rearrange the formula

To solve for \(w\), divide both sides of the formula by \( \ell \). That gives: \(w=A/ \ell\)

Step 2: Substitute provided values

Now we substitute the provided values of \(A\) and \( \ell \) into the rearranged formula. This makes \(w = 36/9\)

Step 3: Evaluate

After substituting, we end up with \(w=36/9\). Carrying out the division results in \(w=4\)

Key Concepts

Area of a Rectangle

The formula for the area of a rectangle is a key concept in mathematics. It is foundational for understanding geometry and can be useful in everyday situations. The area is calculated by multiplying the length (\( \ell \)) by the width (\( w \)). This can be represented as the formula:\[A = \ell \times w\]

• **Length (\(\ell\))**: The longer side of the rectangle.
• **Width (\(w\))**: The shorter side of the rectangle.
• **Area (\(A\))**: The amount of space inside the rectangle.

It's important to visualize how these dimensions work together. Picture a rectangle divided into squares. The total number of squares is the area. This visual can help make sense of why the formula involves multiplication.

Solving for Variables

Solving for a variable means isolating it on one side of an equation. Here, the goal is to express the width (\( w \)) in terms of given values of area and length. When you start with the formula for the area of a rectangle (\( A = \ell w \)), and you know the area and length, you can find the width.

Here's how you solve for \( w \):

• Locate the variable that needs solving. In our case, it's \( w \).
• Rearrange the equation: \( w = \frac{A}{\ell} \).
• Analyze the rearranged formula: It tells you that the width is the area divided by the length.

By rearranging, you essentially reverse the multiplication. This step is crucial when dealing with algebraic equations.

Substitution Method

The substitution method involves replacing variables in an equation with actual numerical values. This method allows you to evaluate or simplify an equation.

In the example, once we have rearranged the formula to \( w = \frac{A}{\ell} \), we use substitution.

• Find the given values for \( A \) and \( \ell \). Here, \( A = 36 \) and \( \ell = 9 \).
• Substitute these values into the formula: \( w = \frac{36}{9} \).
• Calculate the result: Perform the division to find \( w = 4 \).

Using substitution makes complex formulas manageable by converting them into arithmetic problems. This method is essential for solving algebraic expressions with specific values.