Solving Systems Of Linear Equations A Comprehensive Guide

$$\begin{array}{l} 3 y=\frac{3}{2} x+6 \\ \frac{1}{2} y-\frac{1}{4} x=3 \end{array}$$

A. $(3,6)$ B. $(2,1)$ C. no solution D. infinite number of solutions

Introduction to Systems of Linear Equations

In the realm of mathematics, particularly in algebra, a system of linear equations is a collection of two or more linear equations involving the same set of variables. The solution to a system of linear equations is the set of values for the variables that satisfy all equations simultaneously. In simpler terms, it's the point (or set of points) where all the lines represented by the equations intersect. These systems arise in various fields, including engineering, economics, computer science, and physics, making their understanding crucial for problem-solving in real-world scenarios. This article delves into how to solve a given system of linear equations, providing a step-by-step approach to determine the correct solution.

To solve the given system of linear equations, we will employ algebraic manipulation and substitution methods. The system is presented as follows:

$$\begin{array}{l} 3 y=\frac{3}{2} x+6 \\ \frac{1}{2} y-\frac{1}{4} x=3 \end{array}$$

Our objective is to find the values of $x$ and $y$ that satisfy both equations concurrently. We will begin by simplifying each equation and then use substitution or elimination methods to find the values of $x$ and $y$. The solution will reveal whether the lines intersect at a single point, are parallel (no solution), or are the same line (infinite solutions). Understanding these possibilities is essential for interpreting the nature of the system and its implications.

Step-by-Step Solution

To determine the solution to the system of linear equations, let's embark on a detailed, step-by-step journey. This process will not only lead us to the answer but also illuminate the underlying principles of solving such systems. Our focus will be on clarity, precision, and a thorough understanding of each step involved.

Simplify the Equations

Our initial step involves simplifying the given equations to make them more manageable. This often means eliminating fractions and rearranging terms to bring the equations into a standard form. The given system is:

$$\begin{array}{l} 3 y=\frac{3}{2} x+6 \\ \frac{1}{2} y-\frac{1}{4} x=3 \end{array}$$

Let's start with the first equation:

1. $3y = \frac{3}{2}x + 6$

To eliminate the fraction, we can multiply the entire equation by 2:

$2(3y) = 2(\frac{3}{2}x + 6)$

This simplifies to:

$6y = 3x + 12$

Now, let's rearrange the equation to get $x$ and $y$ terms on one side and the constant on the other. We can subtract $3x$ from both sides:

$-3x + 6y = 12$

We can further simplify this equation by dividing all terms by 3:

$-x + 2y = 4$ (Equation 1 Simplified)

Now, let's move on to the second equation:

2. $\frac{1}{2}y - \frac{1}{4}x = 3$

To eliminate the fractions, we can multiply the entire equation by the least common multiple of 2 and 4, which is 4:

$4(\frac{1}{2}y - \frac{1}{4}x) = 4(3)$

This simplifies to:

$2y - x = 12$ (Equation 2 Simplified)

Now we have the simplified system of equations:

$$\begin{array}{l} -x + 2y = 4 \\ -x + 2y = 12 \end{array}$$

Analyze the Simplified Equations

Upon examining the simplified equations, we notice something significant. Both equations have the same coefficients for $x$ and $y$ ($-x + 2y$), but they are equal to different constants (4 and 12, respectively). This observation is crucial because it indicates the nature of the system's solution.

To elaborate, let's consider what this means graphically. Each linear equation represents a line on a coordinate plane. The solution to a system of two linear equations is the point where the two lines intersect. However, in this case, we have two lines with the same slope (since the coefficients of $x$ and $y$ are the same) but different y-intercepts (since the constants are different). This means the lines are parallel.

Parallel lines, by definition, never intersect. Therefore, there is no point $(x, y)$ that can satisfy both equations simultaneously. This leads us to an important conclusion about the system of equations.

Determine the Nature of the Solution

Given that the simplified equations represent parallel lines, we can definitively state that the system of linear equations has no solution. This is because there is no point of intersection between the two lines. The equations are inconsistent, meaning they contradict each other.

In mathematical terms, if we were to attempt to solve this system using methods like substitution or elimination, we would arrive at a contradiction. For instance, if we tried to subtract one equation from the other, we would end up with an equation stating that 0 is equal to a non-zero constant, which is impossible.

Therefore, the correct answer is:

C. no solution

Exploring Different Solution Types for Systems of Linear Equations

Understanding that a system of linear equations can have no solution is crucial, but it's equally important to grasp the other possibilities. A system of linear equations, typically involving two equations and two variables, can manifest in three distinct ways:

1. Unique Solution: This occurs when the two lines represented by the equations intersect at exactly one point. The coordinates of this point constitute the unique solution to the system. Graphically, the lines have different slopes and cross each other on the coordinate plane. Algebraically, solving the system through methods like substitution or elimination yields a specific value for each variable.

2. No Solution: As we've seen in the problem above, this happens when the lines are parallel. Parallel lines have the same slope but different y-intercepts, meaning they never intersect. In such cases, there is no pair of values for the variables that can satisfy both equations simultaneously. Attempting to solve the system algebraically leads to a contradiction, such as 0 equaling a non-zero number.

3. Infinite Solutions: This situation arises when the two equations represent the same line. In other words, one equation is a multiple of the other. Graphically, the lines overlap completely. Any point on the line satisfies both equations, resulting in an infinite number of solutions. Algebraically, the equations are dependent, and solving the system leads to an identity, such as 0 = 0, indicating that there are infinitely many solutions.

Methods to Solve Systems of Linear Equations

Several methods are available to solve systems of linear equations, each with its own advantages and applications. Here are three common methods:

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 solved. The value obtained is substituted back into one of the original equations to find the value of the other variable. Substitution is particularly useful when one of the equations is already solved for one variable or can be easily manipulated to do so.

2. Elimination Method (or Addition Method): The elimination method involves manipulating the equations so that the coefficients of one variable are opposites. By adding the equations, one variable is eliminated, leaving a single equation in one variable. This equation is solved, and the value is substituted back into one of the original equations to find the value of the other variable. Elimination is effective when the equations are in standard form (Ax + By = C) and the coefficients of one variable are easily made opposites.

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. If the lines are parallel, there is no solution; if they coincide, there are infinitely many solutions. The graphical method provides a visual representation of the system and its solutions, making it easier to understand the nature of the solutions. However, it may not be as precise as algebraic methods for finding exact solutions.

Real-World Applications of Systems of Linear Equations

Systems of linear equations are not merely abstract mathematical concepts; they are powerful tools with a wide array of real-world applications. Their ability to model relationships between multiple variables makes them indispensable in various fields. Let's explore some key applications:

1. Engineering: In engineering disciplines, systems of linear equations are used extensively for circuit analysis, structural analysis, and control systems. For example, electrical engineers use Kirchhoff's laws, which result in systems of linear equations, to determine the currents and voltages in electrical circuits. Similarly, civil engineers use systems of equations to analyze the forces and stresses in structures like bridges and buildings.

2. Economics: Economists use systems of linear equations to model supply and demand curves, market equilibrium, and macroeconomic models. For instance, the intersection of supply and demand curves, which can be represented by linear equations, determines the equilibrium price and quantity in a market. Input-output models, which are systems of linear equations, are used to analyze the interdependencies between different sectors of an economy.

3. Computer Science: Systems of linear equations are fundamental in computer graphics, image processing, and machine learning. In computer graphics, linear algebra and systems of equations are used for transformations, projections, and rendering 3D objects onto a 2D screen. In machine learning, linear regression models and neural networks often involve solving systems of linear equations to determine the optimal parameters.

4. Physics: Physics relies heavily on systems of linear equations to solve problems in mechanics, electromagnetism, and thermodynamics. For example, Newton's laws of motion can be expressed as systems of equations to analyze the motion of objects under various forces. In electromagnetism, Maxwell's equations, which are a set of partial differential equations, can be approximated as systems of linear equations in certain situations.

5. Operations Research: In operations research, systems of linear equations are used in linear programming, a technique for optimizing a linear objective function subject to linear constraints. Linear programming is applied in various areas, such as resource allocation, scheduling, and transportation planning, to find the best possible solution given certain limitations.

Conclusion

In conclusion, solving systems of linear equations is a fundamental skill in mathematics with broad applications across various disciplines. The system presented in this article exemplifies a case where no solution exists, highlighting the importance of understanding the nature of linear systems. We explored the step-by-step method to simplify and analyze the equations, revealing that they represent parallel lines. Additionally, we discussed the different types of solutions possible—unique, none, and infinite—and the common methods for solving such systems, including substitution, elimination, and graphical approaches. Furthermore, we delved into the real-world applications of systems of linear equations, demonstrating their significance in engineering, economics, computer science, physics, and operations research.

Mastering the techniques to solve systems of linear equations not only enhances mathematical proficiency but also provides valuable tools for problem-solving in diverse fields. By understanding the underlying principles and applications, one can effectively tackle complex problems and make informed decisions in various contexts.