Is (2,-1,3) A Solution? Step-by-Step Verification

Hey guys! Today, we're diving into a super important concept in mathematics: figuring out if a specific point is a solution to a system of equations. Specifically, we're going to check if the ordered triple (2, -1, 3) is a solution to the following system:

 x + y + z = 4
 x - 2y - z = 1
 2x - y - 2z = -1

Think of this like a puzzle! We have three equations and three unknowns (x, y, and z). Our mission, should we choose to accept it (and we do!), is to see if plugging in the values x = 2, y = -1, and z = 3 into each equation makes the equation true. If it does for all the equations, then we've cracked the code – the ordered triple is indeed a solution.

What Does it Mean to be a Solution?

Before we jump into the nitty-gritty, let's clarify what it means for an ordered triple to be a solution to a system of equations. Imagine each equation as representing a plane in 3D space. A solution to the system is a point that lies on all the planes simultaneously. It's where all the planes intersect. So, if our ordered triple (2, -1, 3) is a solution, it means that point exists on all three planes defined by our equations.

Now, let's get our hands dirty with some calculations!

Step-by-Step Verification

The key here is methodical substitution. We'll take our ordered triple (2, -1, 3) and plug the values into each equation, one at a time. We'll then simplify each side of the equation and see if we get a true statement.

Equation 1: x + y + z = 4

Let's substitute x = 2, y = -1, and z = 3 into the first equation:

2 + (-1) + 3 = 4

Now, let's simplify the left-hand side:

2 - 1 + 3 = 4

4 = 4

Awesome! The first equation holds true. This is a good sign, but we're not done yet. The ordered triple needs to satisfy all the equations in the system.

Equation 2: x - 2y - z = 1

Next up, we'll substitute the values into the second equation:

2 - 2(-1) - 3 = 1

Remember the order of operations (PEMDAS/BODMAS)! We need to deal with the multiplication first:

2 + 2 - 3 = 1

Now, let's simplify:

4 - 3 = 1

1 = 1

Fantastic! The second equation also checks out. Two down, one to go!

Equation 3: 2x - y - 2z = -1

Finally, let's plug the values into the third equation:

2(2) - (-1) - 2(3) = -1

Again, let's follow the order of operations:

4 + 1 - 6 = -1

Simplify:

5 - 6 = -1

-1 = -1

Boom! The third equation is also satisfied. This is the moment of truth!

Conclusion: Is (2, -1, 3) a Solution?

Guys, we did it! We've successfully substituted the values x = 2, y = -1, and z = 3 into all three equations in the system, and in each case, the equation held true. This means that the ordered triple (2, -1, 3) is indeed a solution to the system of equations.

So, the correct answer is A: The ordered triple is a solution to the system.

This process of verifying solutions is fundamental in algebra and beyond. It helps us understand the relationships between equations and the points that satisfy them. You'll encounter this concept in various contexts, from solving linear systems to dealing with more complex mathematical models.

Why is This Important?

Understanding how to determine if a point is a solution to a system of equations is a crucial skill in mathematics for several reasons. It forms the bedrock for more advanced topics and has practical applications in various fields. Let's delve deeper into why this skill is so important:

Foundation for Advanced Math

This concept is a building block for more complex mathematical concepts. When you progress to topics like linear algebra, calculus, and differential equations, you'll encounter systems of equations and the need to find or verify their solutions. Understanding the basics now will make these advanced topics much easier to grasp later. Think of it as learning the alphabet before you can read a book – you need this foundational knowledge to succeed in more advanced mathematical studies.

Problem Solving Skills

Verifying solutions helps you develop crucial problem-solving skills. It teaches you how to methodically approach a problem, break it down into smaller steps, and apply the correct procedures to reach a solution. This methodical approach is not just valuable in mathematics but also in many other areas of life. By practicing this skill, you become a more confident and effective problem solver.

Graphical Interpretation

As we discussed earlier, systems of equations can be visualized graphically. In the case of three variables, each equation represents a plane in 3D space. The solution to the system is the point where all the planes intersect. Verifying a solution algebraically reinforces this graphical interpretation. It helps you connect the abstract algebraic equations to a concrete geometric representation. This connection between algebra and geometry is a powerful tool for understanding mathematical concepts.

Applications in Real-World Scenarios

Systems of equations are used to model various real-world situations. From engineering and physics to economics and computer science, these systems help us analyze and solve problems. For instance, in engineering, they can be used to design structures or circuits. In economics, they can model supply and demand. In computer graphics, they are used to represent 3D objects. Knowing how to determine if a point is a solution is essential for validating the results of these models.

Error Detection

Verifying solutions is also a great way to catch errors. If you've gone through the process of solving a system of equations and you get a potential solution, plugging it back into the original equations can reveal if you made a mistake along the way. If the solution doesn't satisfy all the equations, you know there's an error somewhere in your calculations, and you can go back and find it. This self-checking mechanism is an invaluable skill in any problem-solving scenario.

Logical Reasoning

The process of substituting values and checking if they satisfy equations strengthens your logical reasoning skills. You're essentially engaging in deductive reasoning – starting with the equations and the potential solution, and then logically deducing whether the solution is valid. This type of logical thinking is crucial for success in mathematics and beyond.

Practice Makes Perfect

Just like any skill, mastering the art of verifying solutions requires practice. The more you practice, the more comfortable you'll become with the process, and the better you'll understand the underlying concepts. So, grab some practice problems, work through them step-by-step, and don't be afraid to make mistakes – that's how we learn! Each time you verify a solution, you're reinforcing your understanding and building your mathematical confidence.

In conclusion, determining if an ordered triple is a solution to a system of equations is a fundamental skill with far-reaching implications. It's not just about plugging in numbers; it's about understanding the relationship between equations, developing problem-solving skills, and building a solid foundation for future mathematical endeavors. So, keep practicing, keep exploring, and keep those mathematical muscles strong!