Seven Less Than Twice A Number Is 5

Seven Less Than Twice a Number Is 5: A Deep Dive into Problem Solving

This seemingly simple math problem, "Seven less than twice a number is 5," serves as a fantastic gateway to understanding algebraic problem-solving. While the solution might seem obvious at first glance, exploring the different approaches and underlying mathematical concepts reveals a wealth of knowledge applicable to far more complex equations. This article will delve into the solution, explore variations of the problem, and discuss the broader mathematical principles involved.

Understanding the Problem: Deconstructing the Sentence

Before jumping into the solution, let's carefully break down the sentence: "Seven less than twice a number is 5." Each part holds a crucial piece of the puzzle:

• "A number": This represents our unknown value, typically denoted by a variable like x or n.
• "Twice a number": This translates to 2 times the number, or 2x (if we use x as our variable).
• "Seven less than twice a number": This means we subtract 7 from twice the number, resulting in the expression 2x - 7.
• "is 5": This indicates that the entire expression (2x - 7) equals 5.

Therefore, our problem can be concisely represented as the algebraic equation: 2x - 7 = 5.

Solving the Equation: A Step-by-Step Guide

We'll use the principles of algebra to isolate x and find its value. Here's a step-by-step solution:

1. Add 7 to both sides: This eliminates the -7 from the left side, maintaining the equation's balance. This gives us: 2x - 7 + 7 = 5 + 7 => 2x = 12

2. Divide both sides by 2: This isolates x, giving us the solution: 2x / 2 = 12 / 2 => x = 6

Therefore, the number is 6. We can verify this by substituting 6 back into the original equation: 2(6) - 7 = 12 - 7 = 5. The equation holds true.

Visualizing the Solution: A Geometric Approach

While algebraic manipulation is the standard method, we can also visualize this problem geometrically. Imagine a line segment representing the number x. "Twice a number" means doubling the length of this segment. "Seven less than twice a number" means removing a segment of length 7 from the doubled segment. The remaining length is 5. This visual representation reinforces the concept and can be helpful for beginners to grasp the problem intuitively.

Variations and Extensions: Expanding the Problem

Let's explore some variations and extensions of this basic problem to further solidify our understanding:

Variation 1: Changing the Result

What if the problem stated, "Seven less than twice a number is 11"? The equation would change to 2x - 7 = 11. Following the same steps:

1. Add 7 to both sides: 2x = 18
2. Divide by 2: x = 9

The solution in this case is x = 9.

Variation 2: Introducing Fractions

Let's make it slightly more challenging: "Seven less than three-halves of a number is 5." This translates to (3/2)x - 7 = 5.

1. Add 7 to both sides: (3/2)x = 12
2. Multiply both sides by (2/3) to isolate x: x = 12 * (2/3) = 8

The solution is x = 8.

Variation 3: Negative Numbers

Consider: "Seven less than twice a number is -3." This yields the equation 2x - 7 = -3.

1. Add 7 to both sides: 2x = 4
2. Divide by 2: x = 2

The solution is x = 2. This demonstrates that the problem-solving method remains consistent even with negative numbers.

Variation 4: Word Problems with Multiple Unknowns

Let's consider a more complex word problem: "The sum of two numbers is 15. Seven less than twice the larger number is equal to the smaller number. Find the two numbers."

This problem requires a system of two equations:

• Equation 1: x + y = 15 (where x is the larger number and y is the smaller number)
• Equation 2: 2x - 7 = y

We can solve this system using substitution or elimination. Let's use substitution:

Substitute Equation 1 into Equation 2: 2x - 7 = 15 - x

Add x to both sides: 3x - 7 = 15

Add 7 to both sides: 3x = 22

Divide by 3: x = 22/3

Substitute x back into Equation 1: 22/3 + y = 15

Solve for y: y = 15 - 22/3 = 23/3

Therefore, the two numbers are x = 22/3 and y = 23/3.

Applying the Concepts to Real-World Scenarios

This seemingly simple mathematical concept has numerous real-world applications. Imagine calculating discounts, determining profits, or even solving problems related to speed, distance, and time. The ability to translate word problems into algebraic equations and then solve them is crucial for a wide range of fields.

For example, a retailer might need to determine the original price of an item after a discount. Knowing the discounted price and the percentage discount, they can create an algebraic equation similar to the ones we've solved to find the original price. Similarly, a financial analyst could use these skills to model investment growth or calculate loan payments.

Importance of Practice and Further Learning

Mastering algebraic problem-solving requires consistent practice. Start with simple problems like "Seven less than twice a number is 5" and gradually increase the complexity. Explore more advanced topics like quadratic equations, simultaneous equations, and inequalities. The more you practice, the more comfortable and proficient you will become in translating word problems into equations and finding solutions.

Conclusion: Beyond the Equation

This article explored the seemingly simple equation, "Seven less than twice a number is 5," in depth. We've seen how to solve it algebraically, visualized it geometrically, and explored variations to strengthen our understanding. Beyond the immediate solution, this problem showcases fundamental algebraic concepts that are applicable to a wide range of mathematical and real-world scenarios. Remember that practice and exploration are key to mastering these concepts and unlocking the power of algebraic problem-solving. Continue practicing, challenge yourself with more complex problems, and enjoy the journey of mathematical discovery.