One Number Added To Three Times Another Number Is 24

One Number Added to Three Times Another Number is 24: Exploring the Math and its Applications

This seemingly simple statement, "One number added to three times another number is 24," opens the door to a fascinating exploration of algebra, problem-solving strategies, and the practical applications of mathematical concepts in various fields. Let's delve into this mathematical puzzle, uncovering its solutions, exploring different approaches to solving it, and examining its relevance in real-world scenarios.

Understanding the Problem: Translating Words into Equations

The core of solving this problem lies in translating the given statement into a mathematical equation. Let's represent our unknown numbers using variables:

• Let 'x' represent the first number.
• Let 'y' represent the second number.

Now, let's translate the statement "One number added to three times another number is 24" into an algebraic equation:

x + 3y = 24

This equation forms the foundation of our problem-solving process. Notice how each part of the sentence translates directly into a mathematical symbol:

• "One number" becomes 'x'.
• "Added to" becomes '+'.
• "Three times another number" becomes '3y'.
• "Is 24" becomes '= 24'.

This translation is a crucial step in mathematical modeling, a process used extensively in science, engineering, and economics to represent real-world situations mathematically.

Solving the Equation: Multiple Approaches

This single equation has infinitely many solutions if we consider the possibility of x and y being any real numbers. However, to find specific solutions, we need additional constraints or information. Let's explore some approaches:

1. Solving for One Variable in Terms of the Other

We can rearrange the equation to solve for one variable in terms of the other. Let's solve for 'x':

x = 24 - 3y

This equation tells us that for any value of 'y', we can calculate the corresponding value of 'x' that satisfies the original equation. For example:

• If y = 0, then x = 24
• If y = 1, then x = 21
• If y = 2, then x = 18
• If y = 8, then x = 0

This demonstrates the infinite solution set. Each pair (x, y) represents a point on a straight line when graphed.

Similarly, we can solve for 'y' in terms of 'x':

y = (24 - x) / 3

This provides an alternative way to find pairs of solutions.

2. Graphical Representation

The equation x + 3y = 24 can be easily represented graphically. This offers a visual understanding of the solution set. Plotting the equation on a Cartesian coordinate system reveals a straight line. Every point on this line represents a solution to the equation. The x-intercept (where the line crosses the x-axis, i.e., y=0) is (24, 0), and the y-intercept (where the line crosses the y-axis, i.e., x=0) is (0, 8).

This graphical method provides an intuitive way to visualize the infinite number of solutions. Different points on the line correspond to different pairs of numbers (x, y) that satisfy the equation.

3. Integer Solutions: Adding Constraints

If we restrict our solutions to integers (whole numbers), the number of possible solutions becomes finite. By systematically testing integer values for 'y', we can find corresponding integer values for 'x'. For instance:

• If y = 0, x = 24
• If y = 1, x = 21
• If y = 2, x = 18
• ...and so on until y = 8, x = 0

Beyond y = 8, we obtain negative integer solutions for x.

4. Using Diophantine Equations

This problem could also be considered a linear Diophantine equation, a type of equation where solutions are restricted to integers. While the techniques for solving general Diophantine equations are more complex, in this case, the simple method of substituting integer values for one variable and solving for the other is sufficient.

Real-World Applications: Where This Math Matters

While seemingly abstract, the underlying principles of this simple algebraic equation have broad applicability in various fields:

1. Resource Allocation and Optimization

Imagine a scenario where you have a fixed budget of 24 units (could be dollars, hours, or any other resource). You need to allocate this budget between two activities. One activity costs 1 unit per item, and the other costs 3 units per item. The equation x + 3y = 24 perfectly models this situation, where 'x' represents the number of items from the first activity, and 'y' represents the number of items from the second activity. Finding different solutions provides various possible resource allocation strategies.

2. Mixture Problems in Chemistry

In chemistry, this equation can model mixing solutions of different concentrations. For instance, you might have two solutions, one with a certain concentration and another with three times that concentration. The equation can help determine the amounts of each solution needed to obtain a target concentration.

3. Linear Programming

In operations research and management science, linear programming involves optimizing a linear objective function subject to linear constraints. Equations like x + 3y = 24 can represent one of the constraints in a larger optimization problem, guiding decisions toward the most efficient use of resources.

4. Engineering and Physics

In numerous engineering and physics applications, linear relationships exist between variables. This simple equation represents a fundamental building block in understanding and modeling more complex systems.

5. Game Theory

In certain game theory scenarios, equations of this type can represent the payoff or utility functions for players, where the values of x and y represent strategic choices, and the sum represents the outcome of the interaction.

Expanding the Problem: Introducing Additional Constraints

To move beyond the infinite solution set, we could introduce additional constraints. For example:

• Constraint 1: x > y (One number is greater than the other)
• Constraint 2: x and y are positive integers
• Constraint 3: x + y = 12 (The sum of the two numbers is 12)

Adding constraints transforms the problem into a system of equations, often resulting in a unique solution or a smaller, finite set of solutions. Solving these systems requires techniques like substitution or elimination.

Conclusion: From Simple Equation to Powerful Tool

The seemingly simple equation, x + 3y = 24, illustrates the power of algebraic representation and the importance of problem-solving strategies. While it has infinitely many solutions in its basic form, adding context and constraints can narrow the solution set, revealing practical applications across various disciplines. Understanding this equation and the techniques used to solve it provides a foundational understanding of mathematical modeling and its role in solving real-world problems. The ability to translate word problems into mathematical equations is a critical skill in numerous fields, underlining the relevance and importance of this seemingly simple mathematical statement. Mastering such concepts is vital for success in higher-level mathematics and its application to problem-solving in diverse fields.