No Solution: Linear Combinations In Equation Systems

Understanding systems of equations and their solutions is a fundamental concept in mathematics. When a system of equations has no solution, it indicates a specific relationship between the equations involved. This article explores how to identify linear combinations within such systems, focusing on the given example and providing a comprehensive understanding of the underlying principles.

Identifying Linear Combinations in Systems with No Solution

When we encounter a system of equations that has no solution, it means the lines represented by these equations are parallel and never intersect. This characteristic gives rise to unique properties when we try to combine these equations linearly. To truly grasp this concept, let's break down the original problem and explore the system of equations provided:

The Given System of Equations

The system presented is:

$$\begin{cases} \frac{2}{3}x + \frac{5}{2}y = 15 \\ 4x + 15y = 12 \end{cases}$$

This system is designed to have no solution, which implies that the two lines are parallel. Our goal is to find an equation that represents a linear combination of these two equations, essentially a manipulation that highlights their parallel nature. A linear combination is created by multiplying each equation by a constant and then adding the results.

Understanding Linear Combinations

A linear combination of two equations involves multiplying each equation by a constant and then adding them together. This technique is often used to solve systems of equations, but it also reveals important properties when a system has no solution. In this case, we are looking for a combination that results in a contradiction, which is a hallmark of systems with no solution. The keyword here is contradiction; we need an equation that is inherently false.

Analyzing the Options

The question provides several options, and we need to determine which one could represent a linear combination of the given system. Let's consider the options:

• A. $\frac{4}{3}x = 42$
• B. $0 = -78$
• C. $\frac{15}{2}y = -16$

Option B: 0 = -78

This option, 0 = -78, immediately stands out because it is a contradiction. In mathematics, zero cannot equal -78. This type of statement is the key indicator of a system with no solution when derived through a linear combination. To arrive at this equation, we would manipulate the original equations in such a way that the variables cancel out, leaving us with a false statement.

Why Option B Works

To understand why option B is the correct answer, let’s explore how we might derive it from the original system. The goal is to eliminate both $x$ and $y$ from the equations. We can multiply the first equation by a constant and the second equation by another constant, such that when we add the equations, the variables cancel out.

Consider the system again:

$$\begin{cases} \frac{2}{3}x + \frac{5}{2}y = 15 \\ 4x + 15y = 12 \end{cases}$$

To eliminate $x$, we can multiply the first equation by -6 and the second equation by 1. This gives us:

$$\begin{cases} -6(\frac{2}{3}x + \frac{5}{2}y) = -6(15) \\ 1(4x + 15y) = 1(12) \end{cases}$$

Which simplifies to:

$$\begin{cases} -4x - 15y = -90 \\ 4x + 15y = 12 \end{cases}$$

Adding these two equations, we get:

$(-4x - 15y) + (4x + 15y) = -90 + 12$, which simplifies to $0 = -78$.

This resulting equation is a clear contradiction, demonstrating that the original system has no solution. The lines are parallel, and there is no point of intersection.

Analyzing Other Options

Let's briefly consider why the other options are not representative of a system with no solution:

• Option A: $\frac{4}{3}x = 42$ - This equation can be solved for $x$, indicating a specific value. It doesn't represent a contradiction or the absence of a solution.
• Option C: $\frac{15}{2}y = -16$ - Similarly, this equation can be solved for $y$, showing a specific value. It does not indicate that the system has no solution.

Only option B, 0 = -78, presents a contradiction that signifies the absence of a solution in the system of equations.

Why Systems Have No Solution

To deepen our understanding, it's crucial to discuss why systems of equations might have no solution in the first place. This typically occurs when the equations represent parallel lines. Parallel lines have the same slope but different y-intercepts, meaning they will never intersect, and thus, there is no solution that satisfies both equations simultaneously.

Parallel Lines and Slopes

The slope-intercept form of a linear equation, $y = mx + b$, helps illustrate this point. Here, $m$ represents the slope and $b$ represents the y-intercept. If two lines have the same slope ($m$) but different y-intercepts ($b$), they are parallel. Let's examine our original equations in this context.

First, rewrite the equations in slope-intercept form:

1. $\frac{2}{3}x + \frac{5}{2}y = 15$

   • Multiply by 6 to eliminate fractions: $4x + 15y = 90$
   • Solve for $y$: $15y = -4x + 90$
   • $y = -\frac{4}{15}x + 6$

2. $4x + 15y = 12$

   • Solve for $y$: $15y = -4x + 12$
   • $y = -\frac{4}{15}x + \frac{4}{5}$

Notice that both equations have the same slope ($-\frac{4}{15}$) but different y-intercepts (6 and $\frac{4}{5}$). This confirms that the lines are parallel and the system has no solution.

Contradictions and No Solution

The contradiction we derived ($0 = -78$) mathematically demonstrates that the system cannot be solved. When we perform valid algebraic manipulations on a system of equations and arrive at a false statement, it signifies that the original system is inconsistent and has no solution. This is a powerful concept in linear algebra and is crucial for understanding the nature of systems of equations.

Practical Implications and Further Exploration

Understanding systems of equations with no solutions has practical implications in various fields, including engineering, economics, and computer science. For instance, in linear programming, identifying systems with no feasible solutions is crucial for optimizing resource allocation. In circuit analysis, inconsistent equations can indicate a problem with the circuit design.

Advanced Topics

For those looking to explore this topic further, consider delving into related concepts such as:

• Inconsistent Systems: Systems of equations that have no solution.
• Consistent Systems: Systems of equations that have at least one solution.
• Linear Dependence: A set of vectors (or equations) where one can be written as a linear combination of the others.
• Matrix Representation: Representing systems of equations using matrices, which provides a more concise and powerful way to analyze them.

By understanding these concepts, you can gain a deeper appreciation for the intricacies of linear algebra and its applications.

Conclusion

In summary, when a system of equations has no solution, it implies the lines are parallel, and a linear combination can lead to a contradiction, such as $0 = -78$. This contradiction is a clear indicator that the system is inconsistent and has no solution. Understanding these principles is essential for solving mathematical problems and applying these concepts in real-world scenarios. Through careful analysis and manipulation of equations, we can uncover the underlying nature of mathematical systems and their solutions.

For further exploration of linear equations and systems, consider visiting Khan Academy's Linear Equations and Graphs for comprehensive lessons and practice exercises.