2 More Than 3 Times A Number Is 17

2 More Than 3 Times a Number is 17: Unraveling the Mathematical Mystery

This seemingly simple statement, "2 more than 3 times a number is 17," hides a fascinating journey into the world of algebra. It's a gateway to understanding how mathematical equations represent real-world problems and how we can use them to find solutions. This article will delve into solving this specific equation, explore the underlying concepts, and demonstrate various approaches to tackling similar problems. We'll also touch on the importance of this type of problem-solving in various fields and its relevance in developing critical thinking skills.

Deconstructing the Problem: From Words to Equation

The beauty of algebra lies in its ability to translate word problems into concise mathematical expressions. Let's break down the statement "2 more than 3 times a number is 17" step-by-step:

• "a number": This represents an unknown value, which we typically denote with a variable, usually 'x'.
• "3 times a number": This translates directly to 3x (3 multiplied by x).
• "2 more than 3 times a number": This means we add 2 to the result of 3x, giving us 3x + 2.
• "is 17": This indicates equality, so we set our expression equal to 17: 3x + 2 = 17.

Therefore, our word problem is elegantly condensed into a single algebraic equation: 3x + 2 = 17.

Solving the Equation: Unveiling the Mystery Number

Now that we have our equation, we can employ various algebraic techniques to solve for 'x'. The goal is to isolate 'x' on one side of the equation to determine its value. Here are two common methods:

Method 1: Subtraction and Division

This method involves systematically eliminating terms from one side of the equation until only 'x' remains.

1. Subtract 2 from both sides: This removes the constant term from the left side. 3x + 2 - 2 = 17 - 2 3x = 15

2. Divide both sides by 3: This isolates 'x'. 3x / 3 = 15 / 3 x = 5

Therefore, the number is 5.

Method 2: Using Inverse Operations

This approach relies on performing the inverse operation to undo the operations performed on 'x'.

1. Identify the operations: The equation shows that 'x' is multiplied by 3 and then 2 is added.
2. Perform inverse operations in reverse order: First, we subtract 2 (the inverse of addition), then we divide by 3 (the inverse of multiplication). 3x + 2 = 17 3x = 17 - 2 (Subtracting 2 from both sides) 3x = 15 x = 15 / 3 (Dividing both sides by 3) x = 5

Again, we arrive at the solution: x = 5.

Verification: Checking Our Solution

It's crucial to verify our solution by substituting the value of 'x' back into the original equation:

3x + 2 = 17 3(5) + 2 = 17 15 + 2 = 17 17 = 17

The equation holds true, confirming that our solution, x = 5, is correct.

Expanding the Horizon: Similar Problem-Solving Techniques

The approach used to solve "2 more than 3 times a number is 17" is applicable to a wide range of algebraic problems. Let's consider some variations:

Example 1: 5 less than twice a number is 11.

1. Translate to an equation: 2x - 5 = 11
2. Solve: 2x = 16 x = 8

Example 2: The sum of a number and its double is 21.

1. Translate to an equation: x + 2x = 21
2. Solve: 3x = 21 x = 7

Example 3: A number multiplied by 4 and then increased by 7 is 39.

1. Translate to an equation: 4x + 7 = 39
2. Solve: 4x = 32 x = 8

Real-World Applications: Beyond the Classroom

While seemingly abstract, the ability to translate word problems into equations and solve them is crucial in numerous real-world scenarios:

• Finance: Calculating interest, determining loan repayments, analyzing investment returns.
• Engineering: Designing structures, calculating forces, analyzing systems.
• Physics: Formulating equations of motion, calculating energy, solving for unknown variables.
• Computer Science: Developing algorithms, creating software, solving optimization problems.
• Everyday Life: Calculating grocery bills, determining distances, managing budgets.

Developing Critical Thinking: The Importance of Problem-Solving

Solving algebraic equations like "2 more than 3 times a number is 17" isn't merely about finding a numerical answer; it cultivates essential critical thinking skills:

• Analytical Skills: Breaking down complex problems into smaller, manageable parts.
• Logical Reasoning: Applying logical steps to arrive at a solution.
• Problem-Solving Strategies: Developing different approaches to tackling various challenges.
• Attention to Detail: Ensuring accuracy in calculations and steps.
• Abstract Thinking: Understanding and manipulating abstract concepts and symbols.

Conclusion: A Foundation for Future Learning

The simple equation arising from the statement "2 more than 3 times a number is 17" serves as a powerful introduction to the world of algebra. Mastering the techniques involved not only solves this particular problem but provides a solid foundation for tackling more complex mathematical concepts in the future. It highlights the practical applications of algebra in various fields and emphasizes the importance of developing critical thinking skills – skills that are highly valuable in all aspects of life. The seemingly simple problem unlocks a world of possibilities, demonstrating the power of mathematical reasoning and problem-solving.