Problem 135 - Intermediate Algebra

Question

In the following exercises, solve. Football On the first down, the Chargers had the ball on their 25 -yard line. They lost 6 yards on the firstdown play, gained 10 yards on the second-down play, and lost 8 yards on the third-down play. What was the yard line at the end of the third-down play?

Step-by-Step Solution

Answer

The yard line at the end of the third-down play is the 21-yard line.

Step 1: Determine the Initial Position

The initial position of the ball is on the 25-yard line.

Step 2: Calculate the Position After the First-Down Play

The team lost 6 yards in the first down. Subtract 6 yards from the initial position: New position = 25 - 6 New position = 19-yard line.

Step 3: Calculate the Position After the Second-Down Play

The team gained 10 yards in the second down. Add 10 yards to the new position from Step 2: New position = 19 + 10 New position = 29-yard line.

Step 4: Calculate the Position After the Third-Down Play

The team lost 8 yards in the third down. Subtract 8 yards from the new position from Step 3: New position = 29 - 8 New position = 21-yard line.

Key Concepts

solving linear equations

Solving linear equations involves finding the value of a variable that makes the equation true. In the case of our football problem, although it doesn't explicitly state a variable, we are essentially solving a real-life equation involving yard gains and losses. To solve such equations, you need to keep track of each operation step-by-step. Remember to carry out operations like addition and subtraction correctly as they influence the final position. In simpler words, treat every yard change as an operation that affects your starting point (25-yard line).

integer operations

Integer operations are fundamental in solving problems involving gains and losses, like our football example. Here, we use both addition and subtraction. When dealing with integers:
- Adding a positive number moves you forward (gain yards).
- Subtracting a positive number moves you backward (lose yards).
Let's break it down with our steps:
- Start at 25.
- Lose 6 yards: 25 - 6 = 19.
- Gain 10 yards: 19 + 10 = 29.
- Lose 8 yards: 29 - 8 = 21.
Each operation adjusts the team's position on the field, showing how integer operations lead us to the final yard line.

step-by-step solutions

Breaking down problems into step-by-step solutions helps make complex problems manageable. Let's review our steps again:
1. Start from initial yard line: 25.
2. First down: Subtract 6 yards - 25 - 6 = 19.
3. Second down: Add 10 yards - 19 + 10 = 29.
4. Third down: Subtract 8 yards - 29 - 8 = 21.
Each step builds upon the previous, leading to a clear and logical solution. By following steps systematically, you ensure accuracy and understand the problem-solving process better.