Problem 56 - Algebra 1: Concepts and Skills

Question

In Exercises 54–56, match the real-life problem with an equation. Then solve the problem. A. \(x+15=7\) B. \(15-x=7\) C. \(15+7=x\) D. \(x+15=-7\) The temperature rose 15 degrees to \(7^{\circ} \mathrm{F}\). What was the original temperature?

Step-by-Step Solution

Answer

The original temperature was \(-8^{\circ} \mathrm{F}\).

Step 1: Identify the correct equation

From the given equations and the real-life situation, we can see that equation A: \(x+15=7\) is the one that fits the situation. This equation means that 'some number \(x\) increased by 15 equals 7', which suits the conversation about temperature rising by a few degrees.

Step 2: Solve for unknown x

To find the original temperature, we need to solve for \(x\). We do this by subtracting 15 from both sides of the equation \(x + 15 = 7\). This operation gives us \(x = 7 - 15\).

Step 3: Calculate the result

The operation in step 2 simplifies to \(x = -8\), which means the original temperature was \(-8^{\circ} \mathrm{F}\).

Key Concepts

Real-life Applications

Linear equations play an essential role in understanding and solving daily problems. They help us convert real-world situations into mathematical expressions, making it easier to manage and make predictions. For instance, let's take temperature changes. When we hear that the temperature has risen or fallen, we can use linear equations to determine the unknown values. This is just one of many ways linear equations help us interpret our surroundings.

• Budgeting: Keep track of expenses and savings
• Travel calculations: Calculate speeds and distances
• Cooking: Adjust ingredient quantities in recipes

By transforming these scenarios into linear equations, we gain valuable insights and can make more informed decisions.

Solving Equations

Solving equations often involves finding the value of an unknown variable that makes a given equation true. In our problem, this unknown is the original temperature before a rise occurred.To solve the equation, we follow these steps:- **Identify the equation**: Choose the equation that correctly represents the scenario. In this case, it's \(x + 15 = 7\). This represents a temperature increase from an initial value of \(x\) to \(7^{\circ} \mathrm{F}\).- **Isolate the variable**: Perform inverse operations to isolate the unknown variable. Here, subtracting \(15\) enables us to isolate \(x\), resulting in \(x = 7 - 15\).- **Compute the result**: Finish the calculation to identify the unknown value, which is \(-8\) in this case.Understanding these steps allows us to approach similar problems with confidence and continue building mathematical skills.

Temperature Problems

Temperature problems provide a practical context for learning and applying linear equations since changes in temperature are common occurrences observable in daily life. Consider that when a weather forecast indicates a rise or fall in temperature, it directly links to a linear equation where the rate of change impacts the overall value. Interpreting these changes: - Recognize that temperature increase is represented as an addition to the original value - Understand that a decrease would inversely involve subtraction in equations - Associate these mathematical expressions with physical concepts such as heat transfer or weather patterns By solving temperature-related problems, students sharpen their problem-solving skills and develop an intuitive understanding of how math applies to environmental phenomena. This not only deepens mathematical prowess but also enhances critical thinking and analysis.