Problem 76 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve by writing a sum of signed numbers and adding. On three successive plays, a football team lost 15 yards, gained 13 yards, and then lost 4 yards. What was the team's total gain or loss for the three plays?

Step-by-Step Solution

Answer

The team's total loss for the three plays was 6 yards.

Step 1: Identify the plays and their values

The three plays were as follows: loss of 15 yards, gain of 13 yards, and a loss of 4 yards. With losses represented as negative numbers and gains as positive numbers, they can be written as -15, +13, and -4.

Step 2: Sum up the signed numbers

Add these signed values together to find the team's total gain or loss: (-15) + (+13) + (-4) = -6

Step 3: Interpret the Result

The result is -6. In the context of the situation, this indicates that the team lost 6 yards overall across the three plays.

Key Concepts

Integers in Algebra

When dealing with algebra, integers are the foundational building blocks of the number system that encompasses whole numbers and their opposites. In other words, they comprise positive numbers, negative numbers, and zero. They are vital in understanding various algebraic concepts because they help us to describe quantities that have a direction associated with them, such as debt in financial concepts or elevation in topography.

The concept of integers in algebra extends to real-life scenarios, where we often encounter situations that require adding or subtracting values which can be either above or below a certain point of reference - think about temperatures relative to freezing point, or financial gains and losses. Algebra helps us to model and solve these problems by treating these opposites numbers in a systematic way. For instance, in our football example, yards lost are represented by negative integers, and yards gained by positive integers.

Adding Signed Numbers

The addition of signed numbers can initially seem tricky, but with a clear understanding of the rules, it becomes a straightforward process. Here's a simple run-down:

• Positive numbers are added just like normal numbers.
• Negative numbers when added together get more negative.
• When a positive and a negative number are added together, they essentially cancel each other out based on their absolute values.

Visualizing a number line can help in understanding this concept - moving to the right for positive numbers and to the left for negative numbers. Adding signed numbers is a core skill in algebra, and in our football team example, you can see that adding (-15) for a loss, (+13) for a gain, and again (-4) for a loss leads to an overall negative result, indicating a net loss in yards.

This process becomes second nature with practice, much like how in real life we subconsciously add and subtract gains and losses every day, like when we're balancing a budget or keeping score in a game.

Real-World Algebra Applications

Algebra isn't just confined to the pages of textbooks; it has practical applications in various real-world scenarios that we encounter on a daily basis. From calculating the best deals when shopping, to more complex problem-solving in engineering and physics, algebra serves as a critical analytical tool.

In our example of a football game, algebra allows us to quantify the progression of the game in terms of yards gained or lost. In finance, algebra is used to figure out interest rates and optimize investment returns. In the field of medicine, it is used for calculating dosages and interpreting statistical data. Essentially, algebra helps to make informed decisions by providing a method to evaluate and compare various options, outcomes, or changes.

Understanding algebraic concepts like signed number sums lays the groundwork for tackling these everyday challenges. By applying algebraic thinking, we can break down complex problems into manageable parts and solve them systematically.