Solving Systems Of Equations: Consistency, Dependency, And Solutions

Alright, let's dive into the world of systems of equations! We're going to figure out how many solutions these systems have, whether they're consistent or inconsistent, and if they're independent or dependent. Don't worry, it's not as scary as it sounds! We'll break it down step by step, and you'll be a pro in no time. This problem deals with a specific system of equations:

$egin{array}{l}3 x+y=-2 \ 6 x+2 y=10

\end{array}$

Understanding Systems of Equations

So, what exactly is a system of equations? Well, it's just a set of two or more equations that we're trying to solve together. The solution to a system is the set of values for the variables (in this case, x and y) that satisfy all the equations in the system. Think of it like a treasure hunt – you have multiple clues (equations), and you're trying to find the spot where all the clues lead (the solution).

There are several ways to solve a system of equations, like graphing, substitution, or elimination. Each method has its pros and cons, but they all aim to find the same thing: the values of the variables that make all the equations true simultaneously. For instance, in our provided example, the solution must satisfy both 3x + y = -2 and 6x + 2y = 10.

Now, let's talk about the different types of solutions we can encounter. A system can have:

• One solution: This means the lines represented by the equations intersect at a single point. This is the most straightforward scenario.
• Infinitely many solutions: This happens when the equations represent the same line. Basically, they're the same equation in disguise! Every point on the line is a solution.
• No solution: This occurs when the lines are parallel and never intersect. There's no point that satisfies both equations.

Understanding these possibilities is key to determining the nature of a system of equations. Furthermore, the number of solutions tells us a lot about the relationship between the equations in the system. Keep these in mind as we tackle our example, guys!

Analyzing the Given System

Okay, let's get down to business and analyze the system of equations you gave. We have:

$egin{array}{l}3 x+y=-2 \ 6 x+2 y=10

\end{array}$

Our goal is to figure out the number of solutions and classify the system based on its consistency and dependency. A system is consistent if it has at least one solution; otherwise, it's inconsistent. It's independent if it has a unique solution (one solution), and it's dependent if it has infinitely many solutions (the equations represent the same line).

To start, let's use the method of elimination. Notice that if we multiply the first equation by 2, we get:

$2 * (3x + y) = 2 * (-2)$ which simplifies to $6x + 2y = -4$.

Now we have:

$egin{array}{l}6 x+2 y=-4 \ 6 x+2 y=10

\end{array}$

See the problem here? The left sides of the equations are identical ($6x + 2y$), but the right sides are different (-4 and 10). This means there's no way for these equations to be true simultaneously. If $6x + 2y$ equals -4, it can't also equal 10 at the same time. This is a classic indicator of an inconsistent system.

Determining the Number of Solutions, Consistency, and Dependency

Alright, let's put it all together. We've seen that the system we are working with is:

$egin{array}{l}3 x+y=-2 \ 6 x+2 y=10

\end{array}$

When we tried to manipulate the equations, we found that they contradict each other. One equation tells us that $6x + 2y$ equals -4, while the other says it equals 10. This contradiction means that there is no solution to the system. The lines represented by these equations are parallel and never intersect. Because there is no solution, the system is inconsistent.

Now, let's address dependency. Since the system has no solution, it cannot be dependent. Remember, dependent systems have infinitely many solutions. So, by default, this system is not dependent. Specifically, it is independent because it has no solutions at all. In conclusion, the system has 0 solutions, is inconsistent, and is independent.

Visualizing the Solution

Let's consider how we'd see this if we graphed the equations. The first equation, $3x + y = -2$, can be rearranged to $y = -3x - 2$. This is a line with a slope of -3 and a y-intercept of -2. The second equation, $6x + 2y = 10$, can be rearranged to $y = -3x + 5$. This is also a line, but with the same slope (-3) and a different y-intercept (5).

Because they have the same slope but different y-intercepts, these lines are parallel. Parallel lines never intersect. And since the point of intersection represents the solution to the system, the fact that these lines don't intersect confirms our earlier conclusion that there is no solution.

Graphing is a fantastic way to visually grasp what's happening with systems of equations. It shows you the relationships between the equations and makes it easier to understand the number of solutions and the system's nature. Even if you aren't asked to graph the equations, doing so can provide a valuable visual check on your algebraic work.

Summary and Key Takeaways

Alright, let's recap what we've learned about the system of equations:

$egin{array}{l}3 x+y=-2 \ 6 x+2 y=10

\end{array}$

Here's the breakdown:

• Number of solutions: 0 (because the lines are parallel)
• Consistency: Inconsistent (because there's no solution)
• Dependency: Independent (because there's no solution)

This example perfectly illustrates how different equations can interact and what that means for the solutions. The key is to remember the definitions of consistent/inconsistent and independent/dependent systems. Always consider what each equation means in terms of the variables and how they interact. Keep practicing, and you'll become a pro at solving and classifying systems of equations! You'll be able to quickly determine the nature of a system and its solutions with practice and a solid understanding of the concepts.

Additional Tips and Tricks

Let's arm you with some extra tips to ace these problems:

• Practice, practice, practice! The more systems you solve, the more comfortable you'll become with recognizing patterns and choosing the right method.
• Look for shortcuts. Sometimes, you can spot the relationship between equations right away without going through the whole solving process.
• Double-check your work. Mistakes happen, so always make sure to review your steps and make sure your solution (or lack thereof) makes sense in the context of the problem.
• Get familiar with graphing tools. Software or online calculators can quickly visualize your equations and confirm your algebraic solutions.

And most importantly, remember that systems of equations are just another puzzle to solve. Break it down into smaller steps, stay organized, and you'll be able to conquer any system that comes your way! Now go forth and solve some equations, guys! You got this! Remember to always check your answers to make sure they are correct and make sense in the context of the problem. This can help you catch mistakes and ensure that your solution is valid.