Solution Analysis Of Linear Equations -3x + 4y = 12 And (1/4)x - (1/3)y = 1

In the realm of mathematics, particularly in algebra, systems of linear equations play a pivotal role. These systems, comprising two or more equations with the same set of variables, are fundamental to modeling various real-world scenarios, ranging from resource allocation to circuit analysis. The solutions to these systems represent the points where the lines or planes represented by the equations intersect. This article delves into the process of determining the solution to a system of two linear equations, focusing on the specific case of the equations -3x + 4y = 12 and (1/4)x - (1/3)y = 1. We will explore different methods for solving such systems and then meticulously verify the given solution options to ascertain their validity. Understanding the nature and solutions of linear equation systems is crucial for various applications in science, engineering, and economics, making this a key topic for students and professionals alike.

Methods for Solving Systems of Linear Equations

Before we dive into the specific equations provided, let's first discuss the common methods employed to solve systems of linear equations. There are three primary techniques:

1. Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can then be easily solved. The value of this variable is then substituted back into one of the original equations to find the value of the other variable.

2. Elimination Method: Also known as the addition or subtraction method, this technique aims to eliminate one of the variables by adding or subtracting multiples of the equations. The goal is to manipulate the equations so that the coefficients of one variable are opposites, allowing them to cancel out when the equations are added. This results in a single equation with one variable, which can be solved, and then the result can be substituted back into one of the original equations to find the other variable.

3. Graphical Method: This method involves graphing both equations on the same coordinate plane. The solution to the system is the point(s) where the lines intersect. This method is particularly useful for visualizing the solution and understanding the nature of the system (e.g., whether it has one solution, infinitely many solutions, or no solution). However, it may not always provide precise solutions, especially when the intersection point has non-integer coordinates.

Solving the System: -3x + 4y = 12 and (1/4)x - (1/3)y = 1

Now, let's apply these methods to the given system of equations:

-3x + 4y = 12

(1/4)x - (1/3)y = 1

To simplify the second equation, we can multiply both sides by 12 (the least common multiple of 4 and 3) to eliminate the fractions:

12 * [(1/4)x - (1/3)y] = 12 * 1

This simplifies to:

3x - 4y = 12

Now our system of equations looks like this:

-3x + 4y = 12

3x - 4y = 12

Elimination Method

We can use the elimination method here. Notice that the coefficients of x in the two equations are opposites (-3 and 3), and the coefficients of y are also opposites (4 and -4). If we add the two equations together, we get:

(-3x + 4y) + (3x - 4y) = 12 + 12

This simplifies to:

0 = 24

This result is a contradiction, which means the system of equations has no solution. The lines represented by these equations are parallel and do not intersect. This is a critical point in understanding the nature of linear systems and how their algebraic representation translates into geometric properties. The elimination method, in this case, revealed a fundamental characteristic of the system—its inconsistency.

Graphical Interpretation

To further illustrate why there is no solution, let's rearrange both equations into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

For the first equation, -3x + 4y = 12, we solve for y:

4y = 3x + 12

y = (3/4)x + 3

For the second equation, 3x - 4y = 12, we solve for y:

-4y = -3x + 12

y = (3/4)x - 3

Notice that both equations have the same slope (3/4) but different y-intercepts (3 and -3). This means the lines are parallel, and parallel lines, by definition, never intersect. This graphical confirmation reinforces our algebraic finding that the system has no solution. The concept of parallel lines and their equations is a cornerstone of coordinate geometry, and this example vividly demonstrates its application in solving systems of linear equations.

Verification of Given Solutions

Now, let's analyze the given solution options:

• The system of the equations has exactly one solution at (-8, 3).
• The system of the equations has exactly one solution at (-4, 3).

Since we've already determined that the system has no solution, both of these statements are incorrect. There is no point that satisfies both equations simultaneously.

Conclusion

The system of equations -3x + 4y = 12 and (1/4)x - (1/3)y = 1 (or equivalently, 3x - 4y = 12) has no solution. This was determined using the elimination method, which resulted in a contradiction (0 = 24), and was further confirmed by analyzing the slopes and y-intercepts of the equations when written in slope-intercept form, revealing that the lines are parallel. Therefore, neither of the given solution options is correct. Understanding the different scenarios that can arise when solving systems of equations, such as the case of no solution due to parallel lines, is crucial for a comprehensive grasp of linear algebra and its applications.

Therefore, the correct answer is that neither of the provided statements is true.