Problem 52 - Elementary Algebra Within Reach

Question

Geometry The length of a tennis court is 6 feet more than twice the width (see figure). Find the width of the court when the length is 78 feet.

Step-by-Step Solution

Answer

The width of the tennis court is 36 feet.

Step 1: Creation of the equation

According to the problem statement, it is mentioned that the length of the court is 6 feet more than twice the width. Let's denote the width with the variable \(w\). Therefore we have: length = 2 * \(w\) + 6.

Step 2: Substitute the given value

We know that the length is 78 feet. Substituting, we get: 78 = 2 * \(w\) + 6.

Step 3: Solve for width

To find the width, we need to isolate \(w\). We do this first by subtracting 6 from both sides of the equation: 78 - 6 = 2 * \(w\) or 72 = 2 * \(w\). Then divide by 2 to get: \(w\) = 72 / 2 or \(w\) = 36. Therefore, the width of the tennis court is 36 feet.

Key Concepts

Geometry in Algebra

Geometry and algebra are two branches of mathematics that often intersect, particularly when it comes to solving real-world problems. In our example, understanding the geometric dimensions of a tennis court is the starting point for creating an algebraic equation. The interplay between these two fields allows us to describe spatial relationships with numbers and equations, making abstract problems quantifiable and solvable.

In geometry, we deal with shapes and their properties, such as lengths, widths, and areas. By bringing algebra into play, we are able to set up relationships between these measurements. For instance, when we know the length of the tennis court is related to its width, we can use algebra to express this relationship and find the missing measurement. The key is to translate the given geometric descriptions into algebraic language - in this case, a linear equation.

Solving for a Variable

When given a problem requiring us to find a specific value, such as the width of a tennis court, we're essentially 'solving for a variable.' This process involves manipulating an algebraic equation so that the variable we're solving for is isolated on one side of the equation.

The steps to isolate the variable typically involve performing inverse operations. These are mathematical operations that 'undo' each other, such as addition and subtraction, or multiplication and division. In our example, we start with subtraction to remove the constant term from one side of the equation and then use division to isolate the variable. It's important to remember that whatever operation we perform on one side of the equation, we must also perform on the other side to maintain balance and equality.

Creating Equations

Creating equations is a critical skill in algebra that involves translating words into mathematical symbols and expressions. The first step is to define the variable that we're solving for — in this case, the width of the tennis court, represented by 'w.' Once we have our variable, we use the information provided in the problem statement to set up an equation that models the scenario.

In our tennis court problem, we were told that the length is '6 feet more than twice the width.' This translates algebraically to '2w + 6.' From there, we can use known values (the length of the court) to set up and solve our equation, finding the value of 'w'. It's like solving a puzzle where the pieces are numbers and operations, and creating equations is the first step in putting that puzzle together.