Problem 28 - Prealgebra

Question

The difference between the record high and low temperatures in Charlotte, North Carolina, is \(109^{\circ} \mathrm{F}\). The record low temperature was \(-5^{\circ} \mathrm{F}\). Write and solve an equation to find the record high temperature.

Step-by-Step Solution

Answer

The record high temperature is \(104^{\circ} \mathrm{F}\).

Step 1: Understand the problem statement

We are given that the difference between the record high and low temperatures in Charlotte is \(109^{\circ} \mathrm{F}\). The record low temperature is \(-5^{\circ} \mathrm{F}\). We need to find the record high temperature.

Step 2: Define the variable

Set the variable \( h \) to represent the record high temperature in \(^{\circ} \mathrm{F}\).

Step 3: Write the equation

Since the difference between the high and low temperatures is given as \(109^{\circ} \mathrm{F}\), we can express this mathematically as:\[ h - (-5) = 109 \] This equation represents the difference between the high and low temperatures.

Step 4: Simplify the equation

Simplify the equation by changing the subtraction of a negative to addition:\[ h + 5 = 109 \]

Step 5: Solve for the variable

Subtract \(5\) from both sides of the equation to solve for \( h \):\[ h + 5 - 5 = 109 - 5 \]\[ h = 104 \] This gives us the record high temperature.

Key Concepts

Understanding Equations

An equation is a mathematical statement that asserts the equality of two expressions. In algebra, we often use equations to find unknown numbers, referred to as variables. When we solve equations, we determine the value for this unknown that makes the equation true.

In our problem concerning the high and low temperatures in Charlotte, the equation is formed based on given information about temperature difference. The difference between the record high and low temperature can be modelled with this equation: \[ h - (-5) = 109 \] where:

• \( h \) is the variable for the record high temperature.
• The subtraction of the low temperature from the high gives us the temperature difference.

By creating an equation, we have a powerful tool for understanding and solving the problem systematically.

Temperature Difference Explained

Temperature difference is simply a measure of how much one temperature varies from another. It is an important concept in various scientific fields, especially in weather-related studies.

In this example, Charlotte's record temperature difference is described as the difference between its historic high and low temperatures. This is given by:

• The high temperature minus the record low.
• In terms of the numbers, it was a total of \(109^{\circ} \mathrm{F}\).

Using this difference, we can deduce unknown temperatures if one of the extremes is known. It's essentially finding the gap between the two temperature records.

Effective Problem Solving

The key to effective problem solving in algebra often involves a series of well-defined steps. Let's outline these steps and apply them to the temperature problem:

• **Understand the Problem:** Clearly identify what is given, and what needs to be found. Here, you know the temperature difference and one extreme temperature.
• **Define the Variable:** Choose a symbol, usually \( h \) or \( x \), to stand for the unknown quantity.
• **Translate Words to Mathematics:** Construct an equation that represents the verbal statements of the problem. For Charlotte's temperature, it's \(h - (-5) = 109\).
• **Solve the Equation:** Use algebraic rules to isolate the variable and find its value.
• **Check and Interpret:** Ensure the solution makes sense in the context of the problem. Here, a high of \(104^{\circ} \mathrm{F}\) logically adds up with the low, resulting in the specified difference.

Systematic approach in solving helps in avoiding errors and making the process clearer.

Role of Variables in Algebra

In algebra, variables are symbols that stand in for numbers we don't yet know. They are like placeholders that allow us to write equations and inequalities that can be solved.

• In the temperature problem, \( h \) is the variable representing the unknown high temperature.
• Variables make it easier to handle numbers that aren't directly given, using operations like addition or subtraction to find them.

Variables give flexibility to problems, allowing a general solution that can apply to multiple situations.

By manipulating variables, such as solving for \( h \) in our problem, we gain insights into unseen or unknown values. This manipulation leads us to solutions, making variables crucial elements in problem solving.