Problem 71 - Beginning and Intermediate Algebra

Question

In a series of plays, the San Francisco \(49 \mathrm{ers}\) gain 2 yards, lose 5 yards, and then lose another 20 yards. What is their total gain or loss of yardage?

Step-by-Step Solution

Answer

The 49ers had a total loss of 23 yards.

Step 1: Identify Initial Gain and Loss

List out the yardage gained and lost in the series of plays. The first play is a gain of 2 yards, represented as +2. The second play is a loss of 5 yards, represented as -5. The third play is another loss of 20 yards, represented as -20.

Step 2: Calculate Net Yardage

Add these values together to find the total gain or loss. Start by adding +2 and -5, resulting in -3. Then, add -3 and -20, resulting in -23.

Step 3: Interpret the Result

The net yardage after adding the gains and losses is -23, indicating a total loss of 23 yards for the San Francisco 49ers.

Key Concepts

Addition of Integers

Addition of integers can be thought of as moving forward or backward on the number line, depending on whether the integer is positive or negative. In the context of yardage gained or lost, a positive integer indicates yardage gained, and a negative integer shows yardage lost. When adding integers, it’s important to determine the direction in which you are moving on the number line.

• Positive + Positive: Move forward on the number line.
• Negative + Negative: Move backward on the number line.
• Positive + Negative (or vice versa): Move in the direction of the larger absolute value.

In our exercise example, the positive integer is a gain of 2 yards. When you add this to the losses (negative integers: -5 and -20), it impacts the overall direction of movement on the number line. Begin by considering +2 followed by adding negative numbers:

• +2 + (-5) = -3
• -3 + (-20) = -23

By following this method, you can determine how positive and negative integers add together to result in a final net yardage of -23.

Subtraction of Integers

Subtracting integers often confuses students because it involves finding the difference between two numbers, which can mentally feel like a more challenging task compared to simple addition. However, subtraction of integers can be simplified using the concept of 'adding the opposite.' When we subtract a number, we are actually adding its opposite to the other integer.

• To subtract while dealing with positive and negative integers, convert the subtraction into an addition problem.
• Change the operation from subtraction to addition and then change the sign of the integer being subtracted.

For example, if you need to perform 5 - 8, instead think of it as 5 + (-8). Applying this back to our problem, we see the operations in play whenever we lose yardage, needing to subtract one integer from another. Instead, we add the negative integer to the previous total, seamlessly converting subtraction into addition for simpler processing.

• This method of converting subtraction into addition by using the opposite number and consistently following number line rules helps in overcoming confusion associated with integer subtraction.

Problem-Solving Steps

When tackling integer operation problems like the one in the exercise, following clear steps makes the process manageable and less intimidating. Here’s how you can streamline your approach, using the football yardage example for clarity.

• Identify the Values: Start by writing down each play’s yardage. Label positive or negative depending on whether it’s a gain or loss (e.g., +2, -5, -20).
• Perform the Calculation: Add the integers step by step, starting with the first pair (-5 to +2) and integrating each subsequent play (-20) to see the continuous impact on total yardage.
• Interpret the Results: Once the total calculation is done and you reach -23, interpret what this means in the real-world context (i.e., total loss of 23 yards).

These steps help break down the problem into manageable parts, providing a structured approach to integer operations. Always remember to double-check your steps and ensure each operation correctly follows integer rules. By doing so, you can easily solve similar real-world problems involving gains and losses.