If X 12y And X 6y 90 Then X

Solving the Simultaneous Equations: If x + 12y = x + 6y = 90, then x = ?

This article delves into the solution of a seemingly paradoxical simultaneous equation problem: If x + 12y = x + 6y = 90, then x = ? At first glance, it appears impossible. How can the sum of x and 12y equal the sum of x and 6y? Let's unravel this mathematical puzzle and explore the underlying principles involved. This will cover not just the solution but also related concepts to enhance your understanding of simultaneous equations.

Understanding the Problem: A Seemingly Impossible Equation

The core issue lies in the presentation of the problem. The statement "x + 12y = x + 6y = 90" implies two separate equations:

1. x + 12y = 90
2. x + 6y = 90

These equations, when considered independently, present a contradiction. If we subtract the second equation from the first, we get:

(x + 12y) - (x + 6y) = 90 - 90

This simplifies to:

6y = 0

Therefore, y = 0.

Substituting y = 0 into either equation (1) or (2) gives us:

x + 12(0) = 90 or x + 6(0) = 90

This simplifies to:

x = 90

The Contradiction Resolved: A Closer Look at Simultaneous Equations

The apparent contradiction arises from the assumption that both equations are simultaneously true for the same values of x and y. However, this is not necessarily the case. The problem statement should be interpreted as representing two separate sets of equations. Each set leads to a unique solution for x and y. The initial impression of a contradiction vanishes when we recognize that the equations are independent and not meant to be solved simultaneously as a system.

Solving Simultaneous Equations: A General Approach

Let's review the general methods of solving simultaneous equations. This background will further illuminate the solution to our primary problem and provide a broader understanding of solving such problems.

There are several approaches to solving simultaneous equations:

• Substitution Method: Solve one equation for one variable and substitute that expression into the other equation.
• Elimination Method: Manipulate the equations to eliminate one variable and solve for the other.
• Graphical Method: Plot the equations on a graph; the intersection point represents the solution.
• Matrix Method: Use matrices to solve systems of linear equations, particularly efficient for large systems.

Let's illustrate these methods with examples:

Example 1: Substitution Method

Solve the following simultaneous equations:

1. x + y = 5
2. x - y = 1

From equation (1), we can express x as: x = 5 - y

Substitute this into equation (2):

(5 - y) - y = 1

Solving for y:

5 - 2y = 1 2y = 4 y = 2

Substitute y = 2 back into either equation (1) or (2) to find x:

x + 2 = 5 x = 3

Therefore, the solution is x = 3 and y = 2.

Example 2: Elimination Method

Solve the following simultaneous equations:

1. 2x + y = 7
2. x - y = 2

Add the two equations to eliminate y:

(2x + y) + (x - y) = 7 + 2 3x = 9 x = 3

Substitute x = 3 into either equation (1) or (2) to find y:

2(3) + y = 7 y = 1

Therefore, the solution is x = 3 and y = 1.

Further Exploration: Interpreting the Original Problem

Returning to our original problem, we've already established that the solution, interpreted correctly, yields x = 90 and y = 0.

However, it's crucial to understand why the initial phrasing creates a perception of contradiction. The issue is the implied simultaneous nature of the equations. The statement would be clearer and less ambiguous if it were written as:

"If two sets of equations are defined as follows: Set 1: x + 12y = 90 Set 2: x + 6y = 90

What is the value of x in Set 1 and Set 2 respectively?"

This rephrasing avoids the misconception of a single, contradictory system of equations.

Practical Applications and Extensions

The concepts discussed here extend beyond simple algebraic exercises. Understanding simultaneous equations is fundamental in numerous fields, including:

• Engineering: Solving circuit problems, analyzing structural forces, and modeling dynamic systems.
• Economics: Determining equilibrium prices and quantities in market models.
• Computer Science: Developing algorithms and solving optimization problems.
• Physics: Solving problems related to forces, motion, and energy.
• Chemistry: Calculating reaction rates and equilibrium concentrations.

Conclusion: Understanding the Nuances of Mathematical Problems

This exploration has highlighted the importance of careful interpretation in mathematical problem-solving. The seemingly paradoxical equation "x + 12y = x + 6y = 90" initially presents a contradiction. However, by understanding the independent nature of the equations, we arrive at the correct solution: x = 90 and y = 0. This analysis underscores the importance of precise language in mathematical statements and demonstrates the versatility of various methods for solving simultaneous equations. The ability to accurately interpret and solve such problems is a valuable skill across numerous academic and professional disciplines. The seemingly simple problem provided a valuable opportunity to review fundamental mathematical concepts and their wider applications.