Solving Systems Of Equations: Find X, Y, And Z
Have you ever been faced with a system of equations and felt completely lost? Don't worry, you're not alone! Solving systems of equations is a fundamental skill in mathematics, and once you grasp the techniques, you'll be able to tackle a wide range of problems. In this comprehensive guide, we'll break down the process step by step, using the following system as our example:
3x - y + z = -3
-2x + y - 2z = 8
-4x + 3y - z = 6
Our goal is to find the values of x, y, and z that satisfy all three equations simultaneously. Let's dive in!
Understanding Systems of Equations
Before we start solving, let's make sure we understand what a system of equations is. A system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that make all the equations true. In our case, we have a system of three linear equations with three variables (x, y, and z). This means we're looking for a single set of values for x, y, and z that will make all three equations true at the same time. There are several methods to solve such systems, and we'll explore one of the most common: elimination.
The method of elimination is a powerful technique that involves strategically adding or subtracting equations to eliminate one variable at a time. This simplifies the system, making it easier to solve for the remaining variables. This technique relies on the principle that if you add or subtract equal quantities from both sides of an equation, the equality remains true. When applied carefully to a system of equations, elimination can systematically reduce the complexity of the problem until a solution is found. It's a versatile approach that can be used for systems with two, three, or even more variables, making it a cornerstone of linear algebra and a valuable tool for various mathematical and scientific applications. Understanding the core concept of elimination is essential for anyone looking to master the art of solving systems of equations.
Step 1: Choose a Variable to Eliminate
Looking at our system:
3x - y + z = -3
-2x + y - 2z = 8
-4x + 3y - z = 6
Notice that the y variable has coefficients of -1, 1, and 3. This makes y a good candidate for elimination because it already has opposite signs in the first two equations. This will allow us to easily eliminate y by adding the equations together.
Eliminating a variable is a crucial step in solving systems of equations, and choosing the right variable to eliminate first can significantly simplify the process. When selecting a variable, look for opportunities to easily combine equations and cancel out terms. Variables with coefficients that are already opposites or multiples of each other are ideal candidates. For example, if you have equations with terms like +2y and -2y, adding the equations will immediately eliminate y. Similarly, if coefficients share a common factor, you can multiply one or both equations to make the coefficients match and then subtract to eliminate the variable. Strategic variable selection can minimize the number of steps required and reduce the risk of errors. Mastering this skill is essential for efficient and accurate problem-solving in linear algebra and related fields. It's like choosing the right tool for the job – the right choice can make a difficult task much easier.
Step 2: Eliminate the Chosen Variable from Two Equations
Let's add the first and second equations together:
(3x - y + z) + (-2x + y - 2z) = -3 + 8
Simplifying, we get:
x - z = 5 (Equation 4)
Now, let's eliminate y from the first and third equations. To do this, we'll multiply the first equation by 3:
3 * (3x - y + z) = 3 * (-3)
9x - 3y + 3z = -9
Now we add this modified equation to the third equation:
(9x - 3y + 3z) + (-4x + 3y - z) = -9 + 6
Simplifying, we get:
5x + 2z = -3 (Equation 5)
This step is where the magic of elimination truly shines. By strategically combining equations, we've reduced the complexity of our system. We started with three equations in three variables, and now we have two equations in just two variables (x and z). This smaller system is much easier to handle. The key to this step is careful manipulation of the equations. Multiplying an equation by a constant doesn't change its solution, but it allows us to create matching coefficients for the variable we want to eliminate. Then, adding or subtracting the equations cancels out that variable, leaving us with a simpler equation. This process can be repeated to eliminate more variables until we're left with a single equation in one unknown, which we can easily solve. Mastering this technique is essential for efficiently solving systems of equations and is a fundamental building block for more advanced mathematical concepts.
Step 3: Solve the New System of Equations
We now have a system of two equations with two variables:
x - z = 5 (Equation 4)
5x + 2z = -3 (Equation 5)
We can solve this system using elimination again. Let's multiply Equation 4 by 2:
2 * (x - z) = 2 * 5
2x - 2z = 10
Now, add this to Equation 5:
(2x - 2z) + (5x + 2z) = 10 + (-3)
Simplifying, we get:
7x = 7
x = 1
Now that we have the value of x, we can substitute it back into Equation 4 to find z:
1 - z = 5
-z = 4
z = -4
The process of solving the reduced system of equations is a crucial turning point in the overall solution. It's where the hard work of elimination pays off, and we begin to uncover the values of the variables. This step often involves applying the elimination method again, but this time on a smaller and simpler system. Once we isolate one variable, we can use substitution to find the others. Substitution is a powerful technique where we replace a variable in one equation with its known value, allowing us to solve for another variable. This process can be repeated until all variables have been determined. The key to success in this step is careful and accurate algebraic manipulation. Avoiding errors in arithmetic and paying close attention to signs are essential for arriving at the correct solution. The ability to efficiently solve systems of equations is a valuable skill in many areas of mathematics and science, and this step is a key component of that skill set.
Step 4: Substitute to Find the Remaining Variable
We have x = 1 and z = -4. Now, we can substitute these values into any of the original equations to find y. Let's use the first equation:
3x - y + z = -3
3(1) - y + (-4) = -3
3 - y - 4 = -3
-1 - y = -3
-y = -2
y = 2
Solution
Therefore, the solution to the system of equations is:
x = 1
y = 2
z = -4
We have successfully found the values of x, y, and z that satisfy all three equations in the system. This final step is where we bring all our previous work together to reveal the complete solution. By substituting the values we found for x and z back into one of the original equations, we can solve for the remaining variable, y. This process highlights the interconnectedness of the variables in a system of equations – each variable's value depends on the others. It's like completing a puzzle, where each piece (variable) fits perfectly into place to create the final picture (solution). Careful attention to detail is still crucial at this stage to avoid errors in substitution and calculation. The satisfaction of arriving at the solution after a series of steps is a testament to the power of algebraic techniques and the beauty of mathematical problem-solving.
Verification
It's always a good idea to verify our solution by plugging the values of x, y, and z back into the original equations to make sure they hold true:
- Equation 1: 3(1) - 2 + (-4) = 3 - 2 - 4 = -3 (Correct)
- Equation 2: -2(1) + 2 - 2(-4) = -2 + 2 + 8 = 8 (Correct)
- Equation 3: -4(1) + 3(2) - (-4) = -4 + 6 + 4 = 6 (Correct)
Since the values satisfy all three equations, our solution is correct!
Verifying the solution is a critical step in the problem-solving process, acting as a safety net to catch any potential errors. It involves plugging the calculated values of the variables back into the original equations to ensure they hold true. This process confirms that the solution satisfies all the conditions of the system, providing confidence in the answer. Verification can also help identify mistakes made during the solving process, such as arithmetic errors or incorrect substitutions. By systematically checking each equation, we can ensure the accuracy of our results. This step is not just a formality; it's an integral part of responsible problem-solving and demonstrates a commitment to precision and thoroughness. In many real-world applications, a verified solution is essential for making informed decisions and preventing costly mistakes. It's the final stamp of approval on a job well done.
Conclusion
Solving systems of equations can seem daunting at first, but by breaking it down into manageable steps, you can master this essential skill. Remember to choose a variable to eliminate strategically, carefully combine equations, and always verify your solution. With practice, you'll become more confident and efficient in solving these types of problems.
For further learning and practice, you can explore resources like Khan Academy's Systems of Equations section.
Keep practicing, and you'll become a system-solving pro in no time!
