Identifying Consistent And Dependent Systems Of Equations A Comprehensive Guide

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Hey guys! Let's dive into the world of systems of equations and figure out what it means for a system to be consistent and dependent. We'll break down the concepts and then analyze a few examples to make it crystal clear. So, grab your pencils and let's get started!

Understanding Consistent and Dependent Systems

When we talk about systems of equations, we're essentially looking at two or more equations that involve the same variables. Our goal is often to find the values of those variables that satisfy all the equations simultaneously. Now, here's where the terms 'consistent' and 'dependent' come into play. A consistent system is basically one that has at least one solution. Think of it as the equations playing nice together and having some common ground. On the other hand, a dependent system is a special type of consistent system. It means that the equations in the system are essentially multiples of each other. They represent the same line, just perhaps written in a slightly different form. This leads to an infinite number of solutions because any point that satisfies one equation will automatically satisfy the other.

To really grasp this, let's consider what happens graphically. If we have two equations, each representing a line, a consistent system will have lines that either intersect at a single point (independent and consistent) or are the same line (dependent and consistent). An inconsistent system, however, would have parallel lines that never intersect, indicating no solution. The key takeaway here is that consistent and dependent systems are essentially the same line, offering infinitely many solutions. Now, let’s deep dive to the concept of systems of equations and their solutions. A system of equations is a set of two or more equations that share the same variables. Solving a system of equations means finding values for the variables that make all the equations true simultaneously. There are several methods for solving systems of equations, including graphing, substitution, and elimination. Graphing involves plotting the equations on a coordinate plane and finding the point(s) where the lines intersect. The coordinates of the intersection point(s) represent the solution(s) to the system. Substitution involves solving one equation for one variable and then substituting that expression into the other equation. This results in a single equation with one variable, which can be solved. The solution is then substituted back into one of the original equations to find the value of the other variable. Elimination involves manipulating the equations so that the coefficients of one variable are opposites. Adding the equations together eliminates that variable, resulting in a single equation with one variable. The solution is then substituted back into one of the original equations to find the value of the other variable. The nature of the solutions to a system of equations depends on the relationship between the equations. There are three possibilities: the system has one solution, the system has infinitely many solutions, or the system has no solution. A system with one solution is called an independent system. The lines representing the equations intersect at a single point. A system with infinitely many solutions is called a dependent system. The lines representing the equations are the same, meaning they overlap completely. A system with no solution is called an inconsistent system. The lines representing the equations are parallel and never intersect. Consistent and dependent systems are an important concept in linear algebra and have applications in various fields, including engineering, economics, and computer science. Understanding the different types of systems and their solutions allows us to model and solve real-world problems involving multiple variables and constraints. For example, in economics, systems of equations can be used to model the supply and demand for a product, or to analyze the equilibrium prices in a market. In engineering, systems of equations can be used to analyze the forces and stresses in a structure, or to design control systems. In computer science, systems of equations can be used to solve optimization problems, such as finding the shortest path in a network. In addition to the methods mentioned earlier, there are other techniques for solving systems of equations, such as using matrices and determinants. These methods are particularly useful for solving larger systems with many variables. The study of systems of equations is a fundamental topic in mathematics, with connections to many other areas of the field. From basic algebra to advanced linear algebra, the concepts and techniques related to systems of equations play a crucial role in problem-solving and mathematical modeling. Whether you're a student learning the basics or a professional applying these concepts in your work, understanding systems of equations is an essential skill for anyone working with mathematical models and quantitative data.

Analyzing the Given Systems

Okay, let's put our knowledge to the test! We have three systems of equations to analyze. Our mission is to identify which one is both consistent and dependent. Remember, this means the system must have infinitely many solutions, indicating the equations represent the same line.

System 1:

3x + 2y = 3
6x + 4y = 6

Let's take a closer look at this one. Notice anything interesting? If we multiply the first equation by 2, we get: 2 * (3x + 2y) = 2 * 3, which simplifies to 6x + 4y = 6. Hey! That's exactly the second equation. This tells us that the two equations are just multiples of each other. They represent the same line! So, this system is consistent (because there are solutions) and dependent (because there are infinitely many solutions). To further illustrate why this system is consistent and dependent, let's consider the graphical representation. When we graph the two equations, we'll find that they overlap completely, forming a single line. Every point on this line is a solution to both equations, hence the infinitely many solutions. From an algebraic perspective, we can manipulate the equations to see their equivalence more clearly. If we solve both equations for y, we get:

  • From the first equation: 2y = 3 - 3x => y = (3 - 3x) / 2
  • From the second equation: 4y = 6 - 6x => y = (6 - 6x) / 4 => y = (3 - 3x) / 2

As you can see, both equations yield the same expression for y, confirming that they represent the same line. This system exemplifies a consistent and dependent system, where the equations are essentially the same, leading to an infinite set of solutions. Now, let's shift our focus to System 2 and analyze its characteristics.

System 2:

4x + 6y = 2
4x + 6y = 1

Hmm, this one looks a bit different. We have 4x + 6y on the left side in both equations, but the right sides are different numbers (2 and 1). Can 4x + 6y be equal to both 2 and 1 at the same time? Nope! This is a contradiction. There's no solution that can satisfy both equations. Therefore, this system is inconsistent. In this case, the two equations represent parallel lines. Parallel lines have the same slope but different y-intercepts, meaning they never intersect. Graphically, we would see two lines running side by side, never touching. Algebraically, we can see the inconsistency by attempting to solve the system. If we subtract the second equation from the first, we get:

(4x + 6y) - (4x + 6y) = 2 - 1 0 = 1

This is a clear contradiction, as 0 cannot equal 1. This further confirms that the system has no solution and is therefore inconsistent. Systems like this are important to recognize because they represent situations where there is no common ground between the equations. They often arise in modeling scenarios where there are conflicting constraints or requirements. Now, let's move on to System 3 and see what type of system it represents.

System 3:

2x + 3y = -6
2x + 3y = -4

Similar to System 2, we have the same expression on the left side (2x + 3y) but different constants on the right side (-6 and -4). Just like before, 2x + 3y can't be equal to two different values simultaneously. This system is also inconsistent – there's no solution. This system, like System 2, represents parallel lines. The equations have the same slope but different y-intercepts, preventing them from ever intersecting. We can demonstrate this algebraically by attempting to solve the system. If we subtract the second equation from the first, we get:

(2x + 3y) - (2x + 3y) = -6 - (-4) 0 = -2

Again, we arrive at a contradiction. The equation 0 = -2 is not true, indicating that the system has no solution. Inconsistent systems like this highlight the importance of checking for feasibility when modeling real-world scenarios. They indicate that the constraints or conditions imposed by the equations are incompatible, and no solution can satisfy all of them simultaneously. Understanding these different types of systems allows us to interpret the relationships between equations and their solutions, providing valuable insights into the problems they represent. So, with our analysis complete, we can confidently identify the consistent and dependent system.

The Verdict

Alright, guys, after analyzing all three systems, the winner is...

System 1:

3x + 2y = 3
6x + 4y = 6

This system is consistent and dependent because the equations represent the same line, leading to infinitely many solutions. We nailed it! Remember, the key is to look for equations that are multiples of each other. This indicates they're essentially the same line, just dressed up differently. By understanding the concepts of consistent, inconsistent, and dependent systems, we can effectively analyze and solve a wide range of mathematical problems. Keep practicing, and you'll become a pro at identifying these systems in no time!