Problem 77 - Introductory Algebra for College Students

Question

Solve by writing a sum of signed numbers and adding. \(\begin{array}{lllllllll}\text { A } & \text { football } & \text { team } & \text { started } & \text { with } & \text { the } & \text { football } & \text { at } & \text { the }\end{array}\) 27-yard line, advancing toward the center of the field (the 50-yard line). Four successive plays resulted in a 4 -yard gain, a 2 -yard loss, an 8 -yard gain, and a 12 -yard loss. What was the location of the football at the end of the fourth play?

Step-by-Step Solution

Answer

The final position of the football team is at the 25-yard line, still moving towards the center of the field.

Step 1: Identify the starting position

Identify that the football team starts at the 27-yard line.

Step 2: Determine the result of each play

For each of the four plays, calculate the change in position. A gain implies movement forward, and it is counted as positive. A loss, as it implies movement backward, is counted as negative. As such, the changes in position are: +4 yards, -2 yards, +8 yards, and -12 yards, respectively.

Step 3: Calculate the final position

Add all these position changes to the initial position to calculate the final position. So, the final position is given by \(27 (initial position) + 4 (first play) - 2 (second play) + 8 (third play) - 12 (fourth play).\)

Key Concepts

Algebraic Problem-Solving

At its core, algebraic problem-solving involves finding unknown values or understanding relationships between variables in mathematical expressions. The process relies on a logical progression of steps, much like the method described in the exercise.

In our problem, we began with identifying the initial state – the football team's starting position on the field. Problem-solving in algebra often involves setting the stage with what we know before tackling what we need to discover. Next, understanding the effects of each play is akin to evaluating the terms in an algebraic expression. The +4 yard gain and +8 yard gain are positive terms, symbolizing advancement, while the -2 and -12 yard losses are negative terms, indicating setbacks.

Finally, combining these terms correctly to obtain the solution mirrors the simplification process in algebra. By sequentially accumulating the gains and losses, we arrive at the final position of the football. Just as in algebra, where we combine like terms and simplify expressions to solve for unknowns, we summed the signed numbers to determine the football’s final yard line.

Signed Numbers in Algebra

Signed numbers are the bread and butter of algebra. These numbers include positive and negative integers, and they are essential for representing real-world scenarios like debts and credits, temperature changes, or, as in our exercise, advances and losses on a football field.

Understanding how to work with signed numbers is critical. A positive sign (+) indicates a value that is being added or is above a certain reference point, while a negative sign (-) represents a value that is being subtracted or is below a reference point.

When solving problems involving signed numbers, remember these simple rules:

• Addition of a positive number moves us forward.
• Addition of a negative number, which can be viewed as subtracting a positive number, moves us backward.
• When we combine these operations, we must carefully track the direction each term is taking us.

In the football example, gains move us toward a higher yard line, while losses push us toward a lower one. By maintaining strict adherence to the signs, we ensure accuracy in our final solution.

College Algebra Concepts

The exercise we're tackling not only illustrates basic arithmetic with signed numbers but also touches on foundational college algebra concepts. One such concept is the idea of variables and constants. In this scenario, our constants are the yard numbers that do not change. The yard line where the team begins, 27, is a specific starting point – a constant, much like numbers in an equation.

College algebra also often requires interpreting word problems and translating them into mathematical expressions or equations – a skill that’s critical for success. In this problem, we did just that; we translated four successive football plays into an algebraic expression of signed numbers that needed to be added together.

Additionally, problem-solving in college algebra involves an understanding of sequences and series. While not a sequence in the strictest mathematical sense, the succession of the football plays creates a series of gains and losses that we then had to sum, showing a practical application of this concept. These fundamental concepts establish a groundwork for the more advanced topics of algebra that students will encounter in college.