- Define the total area as the sum of the Blue rug area and the Red rug area: \(A_{total} = 8(x + \frac{1}{2}) + 10(x + \frac{2}{5})\).
- Distribute the multipliers into the parentheses to expand the terms: \(8x + 4 + 10x + 4\).
- Combine the like terms \(8x\) and \(10x\) to get \(18x\), and add the constants \(4\) and \(4\) to get \(8\).
- State the final simplified area expression as \(\boxed{18x + 8}\).
Explanation
- Analysis of the Rug Areas
The goal is to find the total area of the new rug formed by combining a Blue rug and a Red rug. We are given the areas of the individual rugs as expressions in terms of a variable \(x\). The Blue rug has an area of \(8(x + \frac{1}{2})\) square units, and the Red rug has an area of \(10(x + \frac{2}{5})\) square units. To find the total area of the combined rug, we must sum these two expressions:
\[A_{total} = 8\left(x + \frac{1}{2}\right) + 10\left(x + \frac{2}{5}\right)\]
- Expanding the Area Expressions
To simplify the total area expression, we first distribute the coefficients outside the parentheses to the terms inside.
For the Blue rug area:
\[8 \times x + 8 \times \frac{1}{2} = 8x + 4\]
For the Red rug area:
\[10 \times x + 10 \times \frac{2}{5} = 10x + \frac{20}{5} = 10x + 4\]
Now, we add these expanded expressions together:
\[A_{total} = (8x + 4) + (10x + 4)\]
- Simplifying the Combined Area
Next, we combine the like terms to simplify the expression further. We group the terms involving \(x\) and the constant terms:
\[A_{total} = (8x + 10x) + (4 + 4)\]
Summing the coefficients for \(x\) gives \(8 + 10 = 18\), and summing the constants gives \(4 + 4 = 8\).
Thus, the total area of the new rug is:
\[A_{total} = 18x + 8\]
The final simplified area of the new rug is \(\boxed{18x + 8}\) square units.
Examples
Imagine you are renovating a room and decide to combine two rectangular floor mats to cover a specific area. By expressing the area of each mat as a function of its dimensions, you can easily calculate the total footprint of the combined mats. This technique of algebraic simplification is essential in construction, interior design, and manufacturing, where you need to manage multiple components efficiently to determine total resource requirements. Understanding how to add and simplify linear expressions allows you to manage these real-world projects with precision.