Three Linear Equations: Solution Set Is A Line
Let's dive into a fascinating concept in linear algebra. When you're dealing with a system of three linear equations and find that the solution set forms a line, it tells us some important things about the system itself. Specifically, we need to understand whether the system is consistent or inconsistent, and whether it's independent or dependent. So, grab your thinking caps, guys, and let's explore this together!
Understanding the Basics
Before we jump into the specifics, let's quickly recap some foundational ideas. A linear equation is an equation that can be written in the form Ax + By + Cz = D, where A, B, C, and D are constants, and x, y, and z are variables. A system of linear equations is simply a collection of such equations.
- Consistency: A system of equations is said to be consistent if it has at least one solution. This means there's at least one set of values for the variables that satisfies all equations simultaneously. Conversely, an inconsistent system has no solution at all; there's no set of values that can satisfy all equations.
- Dependency: A system of equations is independent if each equation provides unique information that isn't already contained within the other equations. A dependent system, on the other hand, has at least one equation that can be derived from the others. This means one or more equations are redundant.
When we visualize these systems in three-dimensional space, each linear equation represents a plane. The solution to the system is the intersection of these planes. Now, let's consider the scenario where the solution set is a line.
The Case of the Line Solution
When the solution set to a system of three linear equations is a line, it implies a few key characteristics about the system. Imagine three planes intersecting in such a way that their common intersection is a straight line. This can happen in a few different ways, but the overarching theme is that the planes are not entirely independent of each other. Let's break it down.
First, consider the consistency of the system. If the intersection of the three planes is a line, then there are infinitely many points (solutions) that lie on that line. This immediately tells us that the system must be consistent. An inconsistent system, by definition, has no solution, so it couldn't possibly have a line as its solution set. Therefore, the system cannot be inconsistent; it has to be consistent.
Second, think about the dependency of the equations. For the intersection to be a line, at least one of the equations must be dependent on the others. If all three equations were independent, they would typically intersect at a single point (a unique solution) or not intersect at all (no solution). For them to intersect along a line, there must be some redundancy or overlap in the information they provide. In simpler terms, at least one of the planes is not contributing entirely new information; it's somewhat defined by the other two. Consequently, the system must be dependent.
Why Dependency is Key
To truly grasp why dependency is crucial when the solution set is a line, let's consider what would happen if all three equations were independent. In three-dimensional space, three independent planes will typically intersect at a single point. Think of three sheets of paper intersecting at a corner – that corner represents a single, unique solution. If you want these planes to intersect along a line, you need at least one plane to be aligned in such a way that its equation can be derived from the other two. This alignment is what makes the system dependent.
For example, imagine the equations:
- x + y + z = 3
- x - y + z = 1
- 2x + 2z = 4
Notice that the third equation is simply twice the sum of the first two equations minus twice the y terms (which cancel each other out). This dependency is what allows these three planes to intersect along a line, rather than at a single point.
Analyzing the Options
Now that we have a solid understanding of the situation, let's analyze the options provided:
A. The system can be either inconsistent or consistent. B. The system can be either independent or dependent. C. The system...
Based on our discussion, we know that the system must be consistent and must be dependent. Therefore:
- Option A is incorrect because the system cannot be inconsistent.
- Option B is correct because the system must be dependent.
To be absolutely clear: If the solution set to a system of three linear equations is a line, the system is consistent and dependent. Consistent because a line represents an infinite number of solutions, and dependent because at least one equation can be derived from the others.
Examples to Solidify Understanding
Let's walk through a few examples to really nail this concept down. These examples will illustrate how different systems of equations can result in a line as the solution set, and why dependency is always involved.
Example 1: A Simple Dependent System
Consider the following system of equations:
- x + y + z = 1
- 2x + 2y + 2z = 2
- 3x + 3y + 3z = 3
In this case, it's quite obvious that the second and third equations are simply multiples of the first equation. This means they provide no additional information. All three equations represent the same plane in 3D space. The solution set is the entire plane defined by x + y + z = 1. While technically the solution is a plane and not a line, it shows extreme dependency. If we introduced another independent plane intersecting this one, the intersection would be a line.
Example 2: A More Subtle Dependency
Now, let's look at a slightly less obvious example:
- x + y + z = 3
- x - y + z = 1
- x + z = 2
Here, the dependency isn't immediately apparent, but it's there. If you add equations 1 and 2, you get:
(x + y + z) + (x - y + z) = 3 + 1
2x + 2z = 4
Divide by 2:
x + z = 2
This is exactly equation 3! This means equation 3 can be derived from equations 1 and 2, making the system dependent. The solution to this system is the line defined by the intersection of the planes represented by equations 1 and 2 (or, equivalently, equation 1 and 3, or equation 2 and 3).
Example 3: Visualizing the Intersection
Imagine three sheets of paper. If you place two of them on top of each other, they essentially act as one sheet. Now, if you intersect a third sheet with these two, the intersection will be a line (where the third sheet cuts through the first two). This is a visual representation of how a dependent system of three linear equations can have a line as its solution.
Common Mistakes to Avoid
When dealing with systems of linear equations, it's easy to make a few common mistakes. Here are some pitfalls to watch out for:
- Assuming Independence: Always check for dependency before drawing conclusions. Just because equations look different doesn't mean they are independent. Use methods like Gaussian elimination or substitution to determine if one equation can be derived from the others.
- Ignoring Consistency: Remember that a system must be consistent to have a solution set, whether it's a point, a line, or a plane. If you find that the equations contradict each other (e.g., 0 = 1), the system is inconsistent, and there is no solution.
- Misinterpreting Geometric Representations: Visualize the equations as planes in 3D space. This can help you understand how the intersections work and why dependency is necessary for a line to be the solution set.
Conclusion
In summary, when the solution set to a system of three linear equations is a line, the system is necessarily consistent and dependent. This means there are infinitely many solutions lying on the line, and at least one of the equations can be derived from the others. Understanding these concepts is crucial for mastering linear algebra and solving systems of equations effectively. Keep practicing, and you'll become a pro in no time! Remember, guys, linear algebra is all about understanding the relationships between equations and their geometric representations. Keep exploring, and have fun with it!