Problem 56 - Algebra A Combined Function

Question

If the formula for the area of a rectangle, \(A=l \cdot w,\) is solved for \(w,\) then \(w=\frac{A}{l} .\) Use this formula to find \(w\) if area \(A\) is \(\frac{4 x-2}{3}\) square meters and length \(l\) is \(\frac{6 x-3}{5}\) meters. Write \(w\) in simplified form.

Step-by-Step Solution

Answer

The simplified expression for \(w\) is \(\frac{20x - 10}{18x - 9}\).

Step 1: Understand the Formula

The formula for the area of a rectangle is given by \( A = l \cdot w \), where \( A \) is the area, \( l \) is the length, and \( w \) is the width. To find the width \( w \), we rearrange the formula to get \( w = \frac{A}{l} \).

Step 2: Plug in the Given Values

We are given \( A = \frac{4x-2}{3} \) and \( l = \frac{6x-3}{5} \). Substitute these values into the formula for \( w \): \[ w = \frac{\frac{4x-2}{3}}{\frac{6x-3}{5}}. \]

Step 3: Simplifying the Expression

To simplify \( \frac{\frac{4x-2}{3}}{\frac{6x-3}{5}} \), take the reciprocal of the denominator and multiply: \[ w = \frac{4x-2}{3} \times \frac{5}{6x-3}. \] Multiply the numerators and the denominators: \[ w = \frac{(4x-2) \times 5}{3 \times (6x-3)}. \] This becomes: \[ w = \frac{20x - 10}{18x - 9}. \]

Step 4: Find the GCD to Simplify Further

To simplify \( \frac{20x - 10}{18x - 9} \), notice there is no common factor shared across the entire fraction. However, simplify by canceling common factors if needed in context of specific numbers.

Step 5: Simplified Expression

The expression \( w = \frac{20x - 10}{18x - 9} \) is already in its simplest form in terms of variables. Double-check if numerical common factors are cancelable if further context was provided.

Key Concepts

Area of Rectangle

The concept of the area of a rectangle is fundamental in understanding how space is measured for four-sided planar surfaces. The formula for calculating the area of a rectangle is given by the product of its length and width: \[ A = l \times w \] where:

• \( A \) represents the area, which is the entire surface occupied by the rectangle.
• \( l \) is the length, one side of the rectangle.
• \( w \) is the width, the adjacent side of the rectangle.

When rearranging this formula to solve for the width \( w \), it becomes \[ w = \frac{A}{l} \] This rearranged formula is useful when you have a known area and length and need to find the width. Students often perform operations such as this in math to isolate a variable, making it a crucial skill to develop in algebra.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves making them as concise as possible while maintaining their original value. The goal is to reduce confusion and complexity, especially when dealing with fractions or mixed terms.
To simplify:\[ \frac{4x-2}{3} \times \frac{5}{6x-3} \] we followed these steps:

• Multiply the numerators together: \((4x-2) \times 5 = 20x - 10\).
• Multiply the denominators together: \(3 \times (6x-3) = 18x - 9\).
• Combine the results into one fraction: \[ \frac{20x - 10}{18x - 9} \]

In general, always check if you can factor out any constants or terms both in the numerator and denominator. This can lead to a simpler expression if factors are cancelable.

Fractional Expressions

When dealing with fractional expressions, understanding division involving variables is key. These expressions can include variables and constants. Simplifying them often involves reciprocal operations and multiplication rather than straightforward division.
Consider the expression where division of fractions is needed:\[ \frac{\frac{4x-2}{3}}{\frac{6x-3}{5}} \]To simplify:

• Convert the division problem into multiplication by taking the reciprocal of the second fraction: \( \frac{5}{6x-3} \).
• Use that reciprocal to multiply with the first fraction.

This approach helps in avoiding direct division of complex expressions by converting the problem into a multiplication, which is often more straightforward and intuitive.