Problem 110 - Elementary and Intermediate Algebra

Question

The expression \(20,000-3 s\) gives the number of square feet of sod that are left in a field after \(s\) strips have been removed. Suppose a city orders \(7,000\) strips of sod. Evaluate the expression and explain the result. (IMAGE CANT COPY)

Step-by-Step Solution

Answer

Removing 7,000 strips results in -1,000 square feet, meaning there is a deficit of sod.

Step 1: Identify the Given Information

The expression is given as \(20,000 - 3s\), where \(s\) represents the number of strips removed. The city orders \(7,000\) strips of sod, meaning \(s = 7,000\).

Step 2: Substitute the Value of s

Substitute \(s = 7,000\) into the expression \(20,000 - 3s\). The expression becomes \(20,000 - 3 \times 7,000\).

Step 3: Simplify the Equation

First, calculate \(3 \times 7,000\), which equals \(21,000\). Replace \(3 \times 7,000\) with \(21,000\) in the expression. So, the expression now is \(20,000 - 21,000\).

Step 4: Calculate the Remaining Sod

Subtract \(21,000\) from \(20,000\): \(20,000 - 21,000 = -1,000\). This result indicates the situation after the sod stripes are removed.

Step 5: Analyze the Result

The result \(-1,000\) square feet means that removing \(7,000\) strips results in more sod being removed than was available initially, suggesting a deficit of sod. This indicates that the city ordered more strips than can be provided by the available sod.

Key Concepts

Evaluating Expressions

Evaluating expressions is a fundamental concept in algebra that involves calculating the value of an algebraic expression by following specific operations. To evaluate an expression like \(20,000 - 3s\), we need to know the value of the variable \(s\).
In our example, the expression represents the remaining square feet of sod after removing some strips. The city has ordered \(7,000\) strips, making the value of \(s\) equal to \(7,000\).
To evaluate the expression, follow these steps:

• Substitute the value of the variable \(s\) into the expression: \(20,000 - 3 \times 7,000\).
• Perform the multiplication: \(3 \times 7,000 = 21,000\).
• Subtract the result from \(20,000\): \(20,000 - 21,000 = -1,000\).

This process reveals that after removing \(7,000\) strips, a deficit of \(-1,000\) square feet results, meaning more sod was removed than initially available.

Substitution Method

The substitution method is a process used in algebra to replace variables with known values in expressions or equations. This technique simplifies complex problems into more manageable ones, allowing us to find specific solutions.
In this exercise, substitution is used to replace the variable \(s\) in the expression \(20,000 - 3s\) with the given number of strips, \(7,000\).
Here’s how substitution works in this context:

• Identify the variable \(s\) in the expression to substitute with the given value, which is \(7,000\).
• Plug in the value: \(20,000 - 3 \times 7,000\).
• Complete any calculations made possible by the substitution: \(3 \times 7,000 = 21,000\) and then \(20,000 - 21,000\).

Substitution helps clarify situations where algebraic expressions represent real-world problems, converting abstract mathematical relationships into understandable numbers.

Problem Solving in Algebra

Problem-solving in algebra involves applying mathematical principles to understand and solve real-world issues. The example of sod strips illustrates how algebra can model practical situations, revealing potential problems before they occur.
In the case of this exercise, we used an algebraic expression to determine the amount of sod left after removing a certain number of strips. Here’s how problem solving was applied:

• Translate the practical situation into an algebraic expression: \(20,000 - 3s\), where \(s\) is the number of strips ordered.
• Apply substitution and evaluate the expression to find out how the practical outcome (amount of sod left) aligns with the mathematical calculation.
• Analyze the result to determine if it makes sense in the context, such as identifying a deficit of \(-1,000\) square feet when too many strips are ordered.

Algebra helps bridge theoretical math and tangible situations, providing valuable insights that guide decision-making and resource management.