Problem 42 - Elementary Algebra

Question

Translate each sentence to a mathematical statement and then simplify. The quarterback ran the ball three times in last Sunday's football game. He gained 7 yards on one run but lost 3 yards and 8 yards on the other two. What was his total yardage running for the game?

Step-by-Step Solution

Answer

The quarterback's total yardage is -4 yards.

Step 1: Translate the Sentence to Mathematical Expression

To find the total yardage for the quarterback, we need to translate the words into a mathematical expression. He gained 7 yards, which is represented by +7. He lost 3 yards (-3) and 8 yards (-8) on other runs. So, the expression would be: \[ 7 + (-3) + (-8) \]

Step 2: Simplify the Expression

Now, simplify the expression \( 7 + (-3) + (-8) \):1. First, combine the numbers: \( 7 - 3 \) results in 4.2. Next, subtract 8 from the result: \( 4 - 8 \) gives us -4.

Step 3: Final Result

The simplified expression shows that the total yardage for the quarterback is -4. This means he lost 4 yards overall during the game.

Key Concepts

Addition and Subtraction

Addition and subtraction are fundamental mathematical operations. They enable us to combine numbers or quantities and find the difference between them. Understanding these concepts is crucial for solving many real-world problems. In the context of the exercise, addition is used to accumulate the total yardage gained by the quarterback. Subtraction, on the other hand, is used to account for the yards lost.

• **Addition** involves bringing together two or more quantities. For example, gaining 7 yards is represented as adding 7.
• **Subtraction** means removing a certain amount from another. If yards are lost, you subtract them from the total.

When working through an expression like \(7 + (-3) + (-8)\), each step involves performing these operations carefully to arrive at the correct result.
First, adding -3 to 7 involves noticing that you are essentially subtracting 3 from 7, which gives you 4. Then, subtracting 8 from 4 gives the final result of -4. This step-by-step approach helps in clearly understanding how to handle mixed operations of adding and subtracting different values.

Integer Operations

Integer operations include addition, subtraction, multiplication, and division involving whole numbers, which can be positive, negative, or zero. In the exercise, the quarterback's yards are represented using integers to reflect both gains and losses. Working with integer operations means understanding how positive and negative numbers interact.

• Positive integers indicate a gain or increase, while negative integers represent a loss or decrease. For example, gaining 7 yards is +7, while losing 3 yards is -3.
• When adding a negative integer, it's the same as subtracting its absolute value. So, \(7 + (-3)\) is treated like \(7 - 3\).

The challenge in integer operations lies in appropriately managing these positive and negative numbers. Keeping track of the signs is crucial. By treating each negative number as a 'loss', it becomes easier to compute net values as was done with \(7 + (-3) + (-8)\), simplifying to a net loss of 4 yards.

Translating Words to Expressions

Translating words to mathematical expressions involves identifying key numerical information and mathematical operations expressed in a sentence. In the original exercise, the words describe specific actions (gains and losses) which need to be accurately converted into mathematical language.

• **Identify crucial actions** in the problem: He gained and lost certain yards.
• **Translate sentences into numbers and operations**: Gaining 7 yards becomes +7, while losing yards becomes -3 or -8.
• **Combine all these expressions** into a single expression representing the entire situation.

Thinking about language in terms of math requires looking for verbs that usually imply addition or subtraction, such as "gain" or "lose". Understanding this translation process is key because it forms the foundation of setting up and solving the mathematical expression accurately. By breaking down the sentence from the exercise into parts that depict actions on integer quantities, a clear expression \(7 + (-3) + (-8)\) is formed and simplified to find the net result.