Hey guys! Ever stumbled upon a system of equations that looks like a tangled mess? Don't worry, it happens to the best of us. Today, we're going to break down a specific system and solve it together, step by step. We'll focus on clarity and making sure you understand the process so you can tackle similar problems on your own. Let's dive in!
The System We're Tackling
Here's the system of equations we're going to solve:
\begin{cases}
4y + z = -20 \\
-x - 6y - z = 23 \\
-7x - 3y - 6z = 1
\end{cases}
Looks intimidating, right? But trust me, we'll conquer it! Our goal is to find the values of x, y, and z that satisfy all three equations simultaneously. We'll express our solution as an ordered triple (x, y, z).
Step 1: Choose Your Weapon (Elimination or Substitution)
When solving systems of equations, you've got a couple of main tools in your arsenal: elimination and substitution. Both are effective, but sometimes one method is more efficient than the other. In this case, elimination seems like a good fit because we can see that the 'z' terms in the first two equations have opposite signs. This means we can easily eliminate 'z' by adding the equations together. However, we can also use substitution. The goal in this initial step is to determine the most straightforward path to simplify the system and reduce the number of variables in play. By carefully examining the coefficients and the structure of the equations, we can make an informed decision that will minimize the complexity of the subsequent steps. Remember, the optimal method might vary depending on the specific system you're facing, so it's always a good idea to consider both elimination and substitution before committing to a particular strategy. The flexibility to adapt your approach based on the problem's characteristics is a key aspect of mastering system-solving techniques. Mastering both methods will give you flexibility and allow you to choose the most efficient approach for any given system. For this problem, let’s use elimination first to knock out a variable and simplify things.
Step 2: Eliminating 'z' (Our First Target)
Our main goal here is to eliminate one variable to reduce the complexity of the system. We'll start by focusing on the first two equations:
4y + z = -20
-x - 6y - z = 23
Notice how the 'z' terms have opposite signs? This is perfect for elimination! If we add these two equations together, the 'z' terms will cancel out:
(4y + z) + (-x - 6y - z) = -20 + 23
Simplifying this gives us:
-x - 2y = 3
Let's call this new equation Equation (4). This equation only has x and y, which is a step in the right direction!
To eliminate z again, this time using the third equation, we need to manipulate the first equation so that the coefficient of z is a multiple of -6 (the coefficient of z in the third equation). We multiply the first equation by 6:
6 * (4y + z) = 6 * (-20)
24y + 6z = -120
Now, add this modified equation to the third equation:
(24y + 6z) + (-7x - 3y - 6z) = -120 + 1
Simplifying this, we get:
-7x + 21y = -119
Divide the entire equation by -7 to simplify:
x - 3y = 17
Let's call this Equation (5).
By strategically eliminating z, we've managed to distill the original system of three equations with three variables into a more manageable system of two equations with just two variables, x and y. This reduction in complexity is a crucial step in solving systems of equations. By focusing on eliminating one variable at a time, we make the problem more approachable and solvable. Remember, patience and methodical execution are key when tackling these types of problems.
Step 3: Solving for 'x' and 'y' (Two Variables, Two Equations)
Now we have a simpler system with two equations and two variables:
\begin{cases}
-x - 2y = 3 \text{ (Equation 4)}\\
x - 3y = 17 \text{ (Equation 5)}
\end{cases}
Again, elimination looks like a good choice! Notice that the x terms have opposite signs. If we add Equation (4) and Equation (5) together, the x terms will cancel out:
(-x - 2y) + (x - 3y) = 3 + 17
Simplifying, we get:
-5y = 20
Now, we can solve for y by dividing both sides by -5:
y = -4
Awesome! We've found the value of y. Now, we can substitute this value back into either Equation (4) or Equation (5) to solve for x. Let's use Equation (5) (it looks a bit simpler):
x - 3(-4) = 17
x + 12 = 17
Subtracting 12 from both sides, we get:
x = 5
Alright! We've found both x and y. It's like solving a puzzle, piece by piece. With x and y determined, we're now in a prime position to tackle the final variable, z. This methodical approach, where we strategically eliminate variables and solve for the remaining ones, is the essence of solving systems of equations. By breaking down the problem into smaller, manageable steps, we can avoid feeling overwhelmed and maintain a clear path towards the solution. Each variable we solve brings us closer to completing the puzzle.
Step 4: Finding 'z' (The Final Piece)
We've got x = 5 and y = -4. Now we need to find z. We can substitute these values into any of the original three equations. Let's use the first equation, since it looks the simplest:
4y + z = -20
Substituting y = -4, we get:
4(-4) + z = -20
-16 + z = -20
Adding 16 to both sides, we get:
z = -4
Fantastic! We've found z = -4. We've solved for all three variables!
Step 5: The Solution (Ordered Triple)
We've found x = 5, y = -4, and z = -4. So, our solution as an ordered triple is:
(5, -4, -4)
This means that the point (5, -4, -4) is the only point that lies on the intersection of all three planes represented by the original equations. It's the unique solution that satisfies all three conditions simultaneously. To be absolutely sure of our answer, we can always plug these values back into the original equations to verify that they hold true. This final check is a good practice to catch any potential errors and ensure that we've indeed found the correct solution. Solving systems of equations is like navigating a maze; with each step, we get closer to the center, and the ordered triple represents that center point, the solution we've been searching for. Double-checking your work is always a pro move!
Step 6: Verification (Just to be Extra Sure)
Let's plug our solution (5, -4, -4) back into the original equations to make sure it works:
- Equation 1: 4y + z = -20
- 4(-4) + (-4) = -16 - 4 = -20 (Correct!)
- Equation 2: -x - 6y - z = 23
- -5 - 6(-4) - (-4) = -5 + 24 + 4 = 23 (Correct!)
- Equation 3: -7x - 3y - 6z = 1
- -7(5) - 3(-4) - 6(-4) = -35 + 12 + 24 = 1 (Correct!)
Our solution checks out! We've successfully solved the system of equations.
Conclusion: You Did It!
Solving systems of equations can seem daunting at first, but by breaking it down into smaller steps, using elimination or substitution, and verifying our solution, we can conquer even the trickiest problems. Remember, practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time! The key takeaway here is the systematic approach: identify a strategy, execute it methodically, and verify your results. This problem-solving mindset extends far beyond mathematics and is a valuable skill in many areas of life. So, congratulations on mastering this system of equations, and keep up the great work!
Key Points to Remember:
- Elimination: Add or subtract equations to eliminate a variable.
- Substitution: Solve one equation for one variable and substitute it into another equation.
- Verification: Always plug your solution back into the original equations to check your work.
Now go out there and solve some more systems!