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

In this comprehensive article, we will delve into the intricacies of solving systems of linear equations, focusing specifically on the equations -3x + 4y = 12 and (1/4)x - (1/3)y = 1. Understanding how to solve such systems is a fundamental skill in algebra, with applications spanning various fields, including engineering, economics, and computer science. We will explore different methods for solving these equations, analyze the nature of the solutions, and address common pitfalls. Our primary goal is to determine which statement accurately describes the solution to this particular system of equations, examining options such as whether the system has a unique solution at points like (-8, 3) or (-4, 3). Through a step-by-step approach, we aim to provide a clear and detailed explanation that enhances your understanding of linear systems and their solutions. This article will not only guide you through the process but also equip you with the knowledge to tackle similar problems with confidence. Mastering these techniques is crucial for anyone looking to excel in mathematics and related disciplines.

Understanding the Equations

Before diving into solving the system, let's first break down each equation individually. The first equation, -3x + 4y = 12, represents a linear relationship between x and y. This means that if we were to graph this equation, it would form a straight line on the coordinate plane. The coefficients -3 and 4 determine the slope and orientation of the line, while the constant 12 affects its position on the plane. Understanding these coefficients is key to visualizing and manipulating the equation effectively. For instance, we can rearrange the equation to solve for y in terms of x, which gives us y = (3/4)x + 3. This form, known as the slope-intercept form, clearly shows that the slope of the line is 3/4 and the y-intercept is 3. Such transformations are not merely algebraic manipulations; they provide deeper insights into the geometrical representation of the equation. Similarly, the second equation, (1/4)x - (1/3)y = 1, also represents a linear relationship. However, the fractional coefficients might initially appear daunting. To simplify this equation, we can multiply both sides by the least common multiple of the denominators, which is 12. This transforms the equation into 3x - 4y = 12, making it easier to work with. Now, we have two equations that look structurally similar, which can simplify the solving process. Analyzing each equation separately allows us to identify potential strategies for solving the system, such as elimination or substitution, and helps us anticipate the nature of the solutions we might encounter. This preliminary understanding is crucial for a systematic and efficient approach to solving systems of equations.

Methods for Solving Systems of Equations

When it comes to solving systems of linear equations, several methods are available, each with its own strengths and suitability depending on the specific equations at hand. In this section, we will explore two primary methods: the substitution method and the elimination method. Understanding these methods and knowing when to apply each is crucial for efficiently solving various systems of equations. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This method is particularly useful when one of the equations can be easily solved for a variable. For example, if we have an equation like y = 2x + 1, substituting this expression for y into another equation will reduce the system to a single equation in one variable, which can then be solved directly. Once we find the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable. This method is intuitive and straightforward, making it a popular choice for many systems of equations. On the other hand, the elimination method (also known as the addition method) involves manipulating the equations so that when they are added together, one of the variables is eliminated. This is typically achieved by multiplying one or both equations by constants so that the coefficients of one variable are opposites. For instance, if we have equations 2x + y = 5 and x - y = 1, adding these equations directly will eliminate y, resulting in 3x = 6. The elimination method is particularly effective when the coefficients of one variable are easily made opposites or when the equations are in a standard form that facilitates this process. Both the substitution and elimination methods are powerful tools for solving systems of equations. The choice of method often depends on the specific structure of the equations and personal preference. By mastering both methods, you will be well-equipped to tackle a wide range of problems involving linear systems.

Applying Elimination Method to -3x+4y=12 and 3x-4y=12

To effectively solve the system of equations -3x + 4y = 12 and 3x - 4y = 12, the elimination method presents a particularly straightforward approach. As we discussed earlier, the elimination method aims to eliminate one variable by adding the equations together. Looking at our system, we can immediately notice that the coefficients of x in the two equations are -3 and 3, respectively, and the coefficients of y are 4 and -4, respectively. This is quite convenient because adding the equations directly will eliminate both x and y simultaneously. Let’s perform this addition:

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

Simplifying the left side, we get:

-3x + 3x + 4y - 4y = 0

And on the right side:

12 + 12 = 24

So, the resulting equation is:

0 = 24

This outcome, 0 = 24, is a contradiction. A contradiction in the context of solving systems of equations indicates that the system has no solution. Graphically, this means that the two lines represented by the equations are parallel and never intersect. In other words, there are no values of x and y that can satisfy both equations simultaneously. This is a crucial insight into the nature of the system. Recognizing contradictions is an important aspect of solving linear systems, as it saves time and prevents the fruitless search for solutions that do not exist. In this case, the direct application of the elimination method not only simplifies the problem but also quickly reveals the fundamental characteristic of the system: its lack of a solution. Understanding why this occurs involves examining the relationship between the lines represented by the equations, which we will delve into further in the subsequent sections. The key takeaway here is that contradictions are not merely algebraic anomalies; they are significant indicators of the system's properties.

Analyzing the Result No Solution

When we encounter a contradiction, such as 0 = 24, while solving a system of equations, it signifies a critical aspect of the system’s nature: the system has no solution. This is a pivotal finding that carries significant implications both algebraically and geometrically. To fully grasp why a contradiction implies no solution, let’s revisit the equations and their graphical interpretations. The equations -3x + 4y = 12 and 3x - 4y = 12 represent two distinct lines in the coordinate plane. When we attempt to solve the system, we are essentially looking for the point(s) where these lines intersect. The coordinates of any intersection point would be the solution(s) to the system, as they would satisfy both equations simultaneously. However, when the equations lead to a contradiction, it means that there is no such point of intersection. This occurs specifically when the lines are parallel. Parallel lines, by definition, never meet, no matter how far they are extended. Algebraically, parallel lines have the same slope but different y-intercepts. To see this in our example, let’s convert 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 can solve for y as follows:

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

So, the slope is 3/4 and the y-intercept is 3. For the second equation, 3x - 4y = 12, we solve for y:

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

Here, the slope is also 3/4, but the y-intercept is -3. As we can see, both lines have the same slope (3/4), indicating they are parallel, but they have different y-intercepts (3 and -3), confirming they are distinct parallel lines. This graphical understanding reinforces the algebraic result. The contradiction arises because we are trying to find a common point that lies on two parallel lines, which is impossible. Therefore, the system of equations has no solution. Recognizing this connection between contradictions, parallel lines, and the absence of solutions is crucial for a comprehensive understanding of linear systems and their behavior.

Evaluating the Given Statements

Now that we have established that the system of equations -3x + 4y = 12 and 3x - 4y = 12 has no solution, we can evaluate the provided statements regarding potential solutions. The statements suggest that the system might have a unique solution at points such as (-8, 3) or (-4, 3). However, our analysis has definitively shown that there are no solutions to this system, as the equations represent parallel lines that never intersect. Therefore, any statement claiming a specific solution, such as a unique solution at (-8, 3) or (-4, 3), must be false. To further solidify this conclusion, we can attempt to substitute these points into the original equations and see if they satisfy both simultaneously. Let’s start with the point (-8, 3). Substituting x = -8 and y = 3 into the first equation, -3x + 4y = 12, we get:

-3(-8) + 4(3) = 24 + 12 = 36

Since 36 ≠ 12, the point (-8, 3) does not satisfy the first equation. Similarly, substituting into the second equation, 3x - 4y = 12, we get:

3(-8) - 4(3) = -24 - 12 = -36

Since -36 ≠ 12, the point (-8, 3) also does not satisfy the second equation. Next, let’s consider the point (-4, 3). Substituting x = -4 and y = 3 into the first equation, -3x + 4y = 12, we get:

-3(-4) + 4(3) = 12 + 12 = 24

Again, 24 ≠ 12, so the point (-4, 3) does not satisfy the first equation. Substituting into the second equation, 3x - 4y = 12, we get:

3(-4) - 4(3) = -12 - 12 = -24

And -24 ≠ 12, so the point (-4, 3) does not satisfy the second equation either. These substitutions confirm our earlier finding that neither of the proposed points is a solution to the system. This empirical verification is a valuable step in problem-solving, as it reinforces the algebraic analysis and ensures a comprehensive understanding of the solution. The key takeaway is that when a system of equations leads to a contradiction, no point can satisfy both equations, and therefore, any claim of a specific solution is incorrect. Understanding this principle is essential for accurately solving and interpreting systems of equations.

Conclusion

In summary, after thoroughly analyzing the system of equations -3x + 4y = 12 and 3x - 4y = 12, we have definitively determined that the system has no solution. This conclusion was reached through the application of the elimination method, which led to a contradiction (0 = 24). This contradiction indicates that the two lines represented by the equations are parallel and do not intersect. Furthermore, we confirmed this by converting the equations to slope-intercept form, demonstrating that they have the same slope but different y-intercepts. We also evaluated the given statements proposing potential solutions at specific points, such as (-8, 3) and (-4, 3), and showed that these points do not satisfy either equation. This comprehensive analysis highlights the importance of understanding the various methods for solving systems of equations, as well as the significance of interpreting the results correctly. A contradiction is not merely an algebraic anomaly but a clear indicator of the system's inherent nature—in this case, the absence of a solution. This understanding is crucial for accurately solving mathematical problems and for applying these concepts in various real-world scenarios. The ability to recognize and interpret such outcomes is a fundamental skill in mathematics and related fields. By mastering these techniques, you can confidently approach and solve a wide range of problems involving linear systems, enhancing your problem-solving capabilities and mathematical acumen. The key takeaway is the importance of a systematic approach, combining algebraic manipulation with geometrical interpretation to gain a complete understanding of the solution.