Problem 12 - A History of Mathematics: An Introduction

Question

Calculate a quantity such that if it is taken two times along with the quantity itself, the sum comes to 9. (problem 25 of the Moscow Mathematical Papyrus)

Step-by-Step Solution

Answer

Answer: The quantity is 3.

Step 1: Combine Like Terms

We have the equation x + 2x = 9. To combine the like terms (the x's), we add the coefficients of x: x + 2x = (1 + 2)x = 3x So, now we have the equation: 3x = 9

Step 2: Isolate the Variable

Now we want to isolate x by dividing both sides of the equation by the coefficient of x, which is 3: 3x = 9 (3x)/3 = 9/3 This simplifies to: x = 3

Step 3: Check the Solution

To check the solution, let's substitute the value of x back into the original equation: x + 2x = 9 (3) + 2(3) = 9 Evaluating this, we get: 3 + 6 = 9 9 = 9 Since both sides of the equation are equal, the solution x = 3 is correct. So, the quantity we're looking for is 3.

Key Concepts

Algebraic Problem Solving

Algebraic problem solving is all about finding unknown values using mathematical equations. In this exercise, the problem from the Moscow Mathematical Papyrus is a classical example of algebra in action. The goal is to find a specific number that, when added to twice itself, equals 9. We begin by setting up an equation, which is the most critical step in algebraic problem solving.

Here, we start with the equation:

• • Create the expression: \(x + 2x = 9\)
  • This equation shows that if you take a number \(x\) and add \(2x\) (which is the number twice more), you reach 9.

By combining the like terms (the \(x\)'s), we simplify the equation to \(3x = 9\). This shows the power of algebra in organizing and clarifying problems, making them easier to solve.

• Solve for \(x\) by dividing both sides by 3 yields \(x = 3\).

Algebraic problem solving involves translating a word problem into a mathematical equation, simplifying it, and using arithmetic operations to find the unknown values. It's crucial for anyone learning math, as it builds foundational thinking skills used in all kinds of real-life scenarios.

Ancient Egyptian Mathematics

Ancient Egyptian mathematics is fascinating for many reasons. The Moscow Mathematical Papyrus, where this problem originates, is a testament to the sophistication of mathematics in ancient Egypt. This papyrus is one of the oldest mathematical texts, and it provides us insight into how ancient Egyptians approached problem-solving.

• Ancient Egyptians mostly used straightforward arithmetic and logical reasoning to tackle mathematical challenges.
• The problems often centered around practical applications, like measuring fields or distributing goods.

Their system was largely additive, meaning they used only sums for computation, which is simpler than the operations we rely on today. They used symbols to denote numbers and had methodologies that allowed them to solve linear equations similar to our example, although they expressed them differently.
This text suggests they had an implicit understanding of algebraic concepts long before formal algebra was developed. The fact that the ancient Egyptian mathematicians composed such comprehensive works tells us that they valued numerical precision and systematic problem-solving, laying the groundwork for future mathematical advancements.

Linear Equations

Linear equations are one of the building blocks of mathematics. A linear equation represents a straight line when it is plotted on a graph, and it's typically in the form \(ax + b = 0\). The exercise from the Moscow Mathematical Papyrus effectively uses linear equations to arrive at the solution.
In the problem \(x + 2x = 9\), the terms can be combined to form the linear equation \(3x = 9\). Solving linear equations usually involves:

• Combining like terms, which simplifies the problem to its most basic form.
• Isolating the variable to one side of the equation. In this case, dividing the entire equation by 3 to find \(x\).
• Finding the solution by performing the math operations, which yields \(x = 3\).

Linear equations are fundamental because they describe many real-world situations. Whether it's physics, economics, or any area requiring systematic analysis, understanding how to work with and solve linear equations is key to navigating all kinds of problem-solving landscapes. This type of equation is not only easy to work with but also crucial for anyone interested in exploring more complex branches of mathematics.