Problem 77 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve by writing a sum of signed numbers and adding. A football team started with the football at the 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 is at the 25-yard line.

Step 1: Identify Starting Position

The football team starts at the 27-yard line. This is the initial value from where changes (either gains or losses) will occur.

Step 2: Represent Gains and Losses as a Sum of Signed Numbers

Here, a 'gain' is represented as a positive number and a 'loss' as a negative number. The four plays resulted in: a 4-yard gain, a 2-yard loss, an 8-yard gain, and a 12-yard loss. So we have the following sum of signed numbers: +4 (gain), -2 (loss), +8 (gain), -12 (loss).

Step 3: Compute Sum of Gains and Losses

Add together the gains and losses: +4 + (-2) + (+8) + (-12) = -2 yards.

Step 4: Calculate Final Position

To find final position, we need to add the result from step 3 to the initial position. Hence, initial position is 27 yards and sum of gains and losses is -2 yards: 27 + (-2) = 25 yards. Thus, after the four plays, the football is on the 25-yard line.

Key Concepts

Algebraic Addition

Understanding algebraic addition is crucial when solving problems that require adding both positive and negative numbers. Imagine we're lining up arrows on a number line: positive arrows point to the right and negative arrows to the left. When we add these arrows together, they might cancel each other out or move the pointer to a new location. To sum these signed numbers effectively, we follow a simple principle: add all the positive numbers together and all the negative numbers together, then combine the two results.

For instance, if a football team gains 4 yards, loses 2 yards, gains another 8 yards, and then loses 12 yards, we first combine the gains (+4 yards, +8 yards) and get a total gain of +12 yards. We then combine the losses (-2 yards, -12 yards) to get a total loss of -14 yards. Finally, we add the total gain to the total loss (+12 + (-14)) which results in -2 yards. This is the algebraic addition applied in practice: converting real-world changes into a mathematical sum of signed numbers.

Positive and Negative Numbers

Dealing with positive and negative numbers can initially seem like navigating through a maze – one false step and you're utterly lost. However, understanding the simple rules associated with these numbers can turn that maze into a straight path.

Positive numbers are your forward movements; in the case of our football team, these represent yards gained. In contrast, negative numbers symbolize backward steps—or yards lost in the context of the game. These numbers also abide by the rules of additive inverses: when you add a number and its opposite (e.g., +5 and -5), they cancel each other out, resulting in zero.

Using signs correctly is crucial, as it determines the direction in which you're moving on the number line. When presented with a scenario involving both types of numbers, always picture your problem on a number line—it's a strategy that adds clarity to the situation at hand.

Word Problem Strategies

In many instances, students find word problems especially challenging because they're a mix of language and mathematics—parsing information is vital. One effective way to tackle these problems is to carefully read through the scenario, identifying and underlining the key numerical information and the operation required.

Next, translate the word problem into mathematical expressions. For example, let's apply this to our football scenario. The 'starting point' is the 27-yard line. The words 'gain' and 'loss' tell us the direction of movement; they are essentially translations for 'add' and 'subtract.' From there, we list the changes in sequence and use sign symbols to reflect their nature (+ for gains, - for losses). Finally, aggregate the signed numbers for a total change and apply it to the starting point to determine the final outcome.

Always check if your answer makes sense within the context of the problem. When we say the football ended up at the 25-yard line after those plays, it's within the realm of possibility for a football game, suggesting that we've tackled the word problem successfully.