Problem 61 - Algebra A Combined Function

Question

Write each algebraic expression described. The area of the Sahara Desert in Africa is 7 times the area of the Gobi Desert in Asia. If the area of the Gobi Desert is \(x\) square miles, express the area of the Sahara Desert as an algebraic expression in \(x\).

Step-by-Step Solution

Answer

The expression is \(7x\).

Step 1: Identify Key Information

First, consider the key information provided in the problem. The problem states that the area of the Sahara Desert is 7 times the area of the Gobi Desert, and the area of the Gobi Desert is given as \(x\) square miles.

Step 2: Understand the Relationship

Realize that the problem is describing a multiplicative relationship where the Sahara Desert's area is 7 times bigger than the Gobi Desert's area. This means that to find the Sahara Desert's area, you must multiply the area of the Gobi Desert by 7.

Step 3: Formulate the Expression

Considering the relationship described, and knowing that the area of the Gobi Desert is \(x\), the area of the Sahara Desert is obtained by multiplying \(x\) by 7. Therefore, the algebraic expression for the area of the Sahara Desert is \(7x\).

Key Concepts

Multiplicative Relationship

In algebra, a multiplicative relationship involves scaling a quantity by a factor to understand its size relative to another quantity. It's a fundamental concept when dealing with expressions like the one in our exercise. Here, the problem states that the Sahara Desert is 7 times the size of the Gobi Desert.
This means that the Sahara's area can be calculated by multiplying the Gobi's area by 7. When you see a word problem like this one, look for terms like "times," which hint at multiplication. Seeing that data expressed in multiplication like "7 times" gives a direct instruction on how to solve the problem.
So, if the relationship had been "9 times," you'd multiply by 9 instead of 7. Understanding these relationships improves your ability to create and manipulate algebraic expressions seamlessly.

Area Calculation

Area calculations involve determining the size of a surface. In our problem, we're asked to calculate the area of a desert using an algebraic expression. Here, the Gobi Desert area is given as a variable, denoted by \( x \).
To find the area of the Sahara Desert, which is 7 times that of the Gobi Desert, a simple multiplication is used. This concept highlights how area relationships can be expressed algebraically. For instance:

• If \( x = 200 \) square miles for the Gobi Desert, the Sahara's area becomes \( 7 \times 200 = 1400 \) square miles.
• If \( x = 100 \) square miles, then Sahara's area would be \( 7 \times 100 = 700 \) square miles.

These calculations not only help comprehend the algebraic expressions but also provide an insight into spatial relationships expressed with variables.

Variables in Equations

Variables represent unknown or changeable values in mathematical equations, acting as placeholders to solve a problem. In the given exercise, the variable \( x \) signifies the Gobi Desert's area in square miles.
By understanding how variables work, it's easier to form expressions to solve bigger problems. The Sahara Desert is given as \( 7x \), meaning the Sahara's area adapts with changes in \( x \). For illustration:

• If \( x \) grows, so does \( 7x \). For example, if \( x \) changes from 100 to 200, \( 7x \) becomes \( 700 \) to \( 1400 \).
• Similarly, if \( x \) decreases, \( 7x \) reduces proportionally.

Variables, therefore, are vital for flexibility in calculations and allow us to explore various scenarios simply by adjusting the value of \( x \). Understanding and using variables effectively is crucial for algebraic problem-solving.