Problem 28 - Elementary Algebra

Question

A number is tripled, then increased by seven. The result is \(48 .\) What is the number?

Step-by-Step Solution

Answer

Answer: The original number is \(\frac{41}{3}\) or approximately 13.67.

Step 1: Define the variables

Let x be the original number we are trying to find.

Step 2: Translate the problem into an equation

We know that the number is tripled, then increased by 7 and the result is 48. We can write this as an equation: 3x + 7 = 48

Step 3: Solve the equation

To find the original number, we need to solve the equation for x. We start by subtracting 7 from both sides: 3x = 41 Now, divide both sides by 3 to solve for x: x = \frac{41}{3}

Step 4: Interpret the solution

The original number is \(\frac{41}{3}\) or approximately 13.67.

Key Concepts

Understanding Equations

Equations are a fundamental part of algebra, forming the foundation for solving problems where unknown values need to be found. In simple terms, an equation is like a balance scale, with expressions on either side of the "=" symbol. The goal is to find the value of the unknown variable that will make both sides equal.
For example, in the exercise provided, the equation is set up to reflect a real-world problem: \(3x + 7 = 48\). Here:

• The left side \(3x + 7\) represents the process described in words (tripling a number and adding seven).
• The right side, 48, is the result of this process.

The solution lies in performing operations that simplify the equation to find the unknown number that satisfies both sides. This typically involves inverse operations like addition or subtraction, and multiplication or division. In our example, subtracting 7 and then dividing by 3 reveals the solution.

Approaching Problem Solving in Algebra

Algebraic problem solving is all about translating a verbal description into a mathematical expression and then solving it. The first step is always to understand what the problem is asking for and what information is provided. The next step is to convert this information into a mathematical form – this is where equations come into play.
The exercise starts with understanding the scenario: "A number is tripled, then increased by seven. The result is 48." To solve it, you follow these steps:

• Identify the unknown, or what you need to find (here, the original number).
• Construct an equation based on the problem description (\(3x + 7 = 48\)).
• Solve the equation systematically to find the unknown value.

Through these steps, you learn how to break down a given problem and find a logical pathway to the solution by using algebraic methods. Effective problem solving involves clear logical thinking and a systematic approach.

The Role of Variables

In algebra, variables are symbols used to represent unknown values. They act as placeholders that can be manipulated following algebraic rules. Commonly used symbols include \(x\), \(y\), and \(z\). Variables allow us to create expressions and equations that can model real-world phenomena.
In the exercise, the variable \(x\) stands for the number we are trying to find. Naming this unknown helps us set up an equation and keep track of what each part of the equation represents.
Variables are very versatile:

• They can represent a single number or a range of possible numbers.
• They can shift between different values depending on the context or constraints given by the problem.

Using variables is essential for abstract thinking in algebra and forms the core of mathematical modeling of physical situations. Understanding variables and how to manipulate them is crucial for mastering algebra.