Problem 45 - Elementary Algebra

Question

Translate each sentence to a mathematical statement and then simplify. The temperature was \(22^{\circ}\) at 6: 00 p.m. and dropped \(26^{\circ}\) by midnight. What was the temperature at midnight?

Step-by-Step Solution

Answer

The temperature at midnight was \(-4^{\circ}\).

Step 1: Define Variables

Let the temperature at 6:00 p.m. be denoted by \( T_6 = 22^{\circ} \). The temperature drop from 6:00 p.m. to midnight is given as \( 26^{\circ} \). We want to find the temperature at midnight, denoted as \( T_m \).

Step 2: Set Up the Mathematical Statement

The problem states that the temperature decreases by \( 26^{\circ} \). This can be represented mathematically as:\[ T_m = T_6 - 26 \]

Step 3: Substitute Values

Substitute the known value of \( T_6 = 22^{\circ} \) into the equation from Step 2:\[ T_m = 22 - 26 \]

Step 4: Simplify the Equation

Now calculate the temperature at midnight by simplifying the equation:\[ T_m = 22 - 26 = -4 \]

Step 5: Interpret the Result

The result \( T_m = -4 \) indicates the temperature at midnight. Therefore, it was \(-4^{\circ}\) at midnight.

Key Concepts

Understanding Temperature Change

Temperature change refers to the difference between two temperature readings at different times. In real-world scenarios like weather forecasting or scientific experiments, it's important to understand how to calculate these changes. In our exercise, the temperature at 6:00 p.m. is given as \(22^{\circ}\). By midnight, this temperature has "dropped" by \(26^{\circ}\). Here, "dropped" indicates a decrease, meaning you subtract the change from the initial temperature. This fundamental concept of finding the difference allows you to determine the temperature at a later time.

Interpreting Mathematical Statements

A mathematical statement is essentially a translation of a word problem into a mathematical expression or equation. This skill is critical in algebra, as it enables you to solve real-world problems mathematically. In the example from the exercise, the problem can be converted into the statement:

• Original temperature: \(T_6 = 22^{\circ}\)
• Temperature drop: \(26^{\circ}\)
• New temperature: \(T_m = T_6 - 26\)

This process involves identifying the initial value, the change (either increase or decrease), and using these elements to find the unknown (the temperature at midnight). Doing this helps clarify the problem and lays the groundwork for finding a solution.

Variable Substitution in Mathematics

Variable substitution is a key algebraic method involving the replacement of a variable by a known value. This allows you to convert an equation into a simpler form that is easier to solve. In the provided exercise, we use substitution to find the temperature at midnight by substituting the known value of the initial temperature \(T_6 = 22^{\circ}\) into the equation:

• Original equation: \(T_m = T_6 - 26\)
• Substitute \(T_6 = 22\)
• Resulting equation: \(T_m = 22 - 26\)

Simplifying this gives you \(T_m = -4\), where \(-4^{\circ}\) is the final temperature. Substitution is critical, enabling you to handle equations with known and unknown variables efficiently.