Question

If the temperature rises \(10^{\circ} \mathrm{F}\), it will be \(-25^{\circ} \mathrm{F}\). What is the temperature now?

Short answer

The initial temperature is -35°F.

Step-by-step solution

Understand the problem

We are given that if the temperature rises by 10°F, it will be -25°F. We need to find the initial temperature.

Set up the equation

Let the initial temperature be denoted as T. According to the problem, if the temperature increases by 10°F, it becomes -25°F. This can be written as: \( T + 10 = -25 \)

Solve for T

To find the initial temperature, isolate T by subtracting 10 from both sides of the equation: \( T + 10 - 10 = -25 - 10 \) Simplifying this, we get: \( T = -35 \)

Key concepts

Temperature Changes

Temperature changes are common in many real-life scenarios, like weather forecasts and climate studies. Understanding how temperature can increase or decrease is crucial in solving word problems involving temperature. In our exercise, it is stated that the temperature will rise by 10°F. To visualize this:

• Imagine the current temperature is represented by a variable, let’s say T.
• If the temperature rises by 10°F, we add 10 to the current temperature T.
• The final temperature is given as -25°F.

This concept sets the stage for forming an equation that we will solve to find the initial temperature.
By grasping how changes in temperature affect an equation, we better understand how to model real-world problems mathematically.

Solving Linear Equations

A linear equation is an equation that makes a straight line when graphed. It often includes variables that we need to solve for. Here, we have a basic linear equation:
\( T + 10 = -25 \)
To solve this linear equation, follow these simple steps:

• Recognize the format of the linear equation: It's in the form of \( T + b = c \) where b and c are constants.
• Our goal is to find the variable T.

Solving linear equations involves simple operations such as addition, subtraction, multiplication, or division. The aim is to isolate the variable on one side of the equation, making it easy to find its value.
This general methodology will help you handle similar problems involving linear equations effectively.

Isolating Variables

Isolating the variable means rearranging the equation so that the variable of interest is by itself on one side of the equation. This is a critical step in solving equations.
For our problem, we need to isolate T in the equation \( T + 10 = -25 \).
Here’s how you do it:

• Start with the given equation: \( T + 10 = -25 \)
• Subtract 10 from both sides of the equation to undo the addition.
• This gives us: \( T + 10 - 10 = -25 - 10 \)
• Simplify the equation: \( T = -35 \)

Now, we have successfully isolated T and found that the initial temperature is -35°F.
Through this process, you learn that isolating variables often involves performing the opposite operation to both sides of the equation. This technique is universally useful across various types of equations.