Solving Systems Of Equations Finding X, Y, And Z
Finding the solutions to a system of equations can seem daunting, but with a systematic approach, it becomes a manageable task. In this article, we'll walk through the process of solving a specific system of equations for the unknown variables x, y, and z. We'll break down each step, providing a clear explanation and ensuring you understand the underlying concepts. Whether you're a student tackling algebra problems or simply looking to refresh your math skills, this guide will empower you to confidently solve similar systems of equations.
The System of Equations
Let's start by examining the system of equations we aim to solve:
x + y = 5
x + z = 14
y + z = 15
Our goal is to determine the values of x, y, and z that satisfy all three equations simultaneously. To achieve this, we'll employ a combination of algebraic techniques, including substitution and elimination.
Step 1: Isolate a Variable
The first step in solving this system is to isolate one variable in terms of the others. Let's choose the first equation, x + y = 5, and solve for x:
x = 5 - y
Now we have an expression for x in terms of y. This is a crucial step as it allows us to substitute this expression into other equations, effectively reducing the number of variables in those equations.
Step 2: Substitute into Another Equation
Next, we'll substitute the expression for x (5 - y) into the second equation, x + z = 14:
(5 - y) + z = 14
Simplifying this equation, we get:
-y + z = 9
Now we have a new equation involving only y and z. This is a significant step forward as we've eliminated x from this equation.
Step 3: Create a System of Two Equations
We now have two equations involving y and z:
-y + z = 9
y + z = 15
This is a system of two equations with two unknowns, which is much easier to solve than our original system. We can use various methods to solve this, such as elimination or substitution. Let's use elimination in this case.
Step 4: Eliminate a Variable
Notice that the y terms in the two equations have opposite signs. This makes elimination a straightforward choice. By adding the two equations together, we can eliminate y:
(-y + z) + (y + z) = 9 + 15
Simplifying, we get:
2z = 24
Now we can solve for z:
z = 12
We've successfully found the value of z!
Step 5: Back-Substitute to Find Other Variables
Now that we know z = 12, we can substitute this value back into one of the equations involving y and z to find y. Let's use the equation y + z = 15:
y + 12 = 15
Solving for y:
y = 3
We've found the value of y!
Finally, we can substitute the value of y (3) back into the equation x = 5 - y to find x:
x = 5 - 3
x = 2
Step 6: Write the Solution
We have now found the values of x, y, and z: x = 2, y = 3, and z = 12. We can write the solution as an ordered triple in the format (x, y, z):
(2, 3, 12)
This ordered triple represents the point where all three equations intersect, meaning it's the solution that satisfies all three equations simultaneously. To check our solution, we can substitute these values back into the original equations and verify that they hold true.
Verification
Let's substitute x = 2, y = 3, and z = 12 into the original equations:
Equation 1: x + y = 5
2 + 3 = 5 (True)
Equation 2: x + z = 14
2 + 12 = 14 (True)
Equation 3: y + z = 15
3 + 12 = 15 (True)
Since all three equations hold true, our solution (2, 3, 12) is correct.
Key Concepts and Strategies
Solving systems of equations is a fundamental skill in algebra, and several key concepts and strategies are involved. Understanding these concepts will empower you to tackle a wide range of problems.
1. Substitution Method
The substitution method involves solving one equation for one variable and then substituting that expression into another equation. This process eliminates one variable, making the system easier to solve. We used this method in Step 1 and Step 2 of our solution.
The power of substitution lies in its ability to simplify complex systems. By expressing one variable in terms of others, we reduce the dimensionality of the problem, allowing us to work with simpler equations. The key to successful substitution is choosing the right equation and variable to isolate, often focusing on those that are easiest to manipulate.
2. Elimination Method
The elimination method involves adding or subtracting equations to eliminate one variable. This is particularly effective when the coefficients of one variable are opposites or can be made opposites by multiplying one or both equations by a constant. We used this method in Step 4 of our solution.
Elimination is a powerful technique when equations are structured in a way that allows for direct cancellation of terms. Identifying opportunities for elimination often involves looking for coefficients that are the same or opposites. By strategically manipulating the equations, we can eliminate variables and simplify the system.
3. Back-Substitution
Once you've found the value of one variable, you can substitute it back into other equations to find the values of the remaining variables. This process is called back-substitution and is crucial for completing the solution. We used this method in Step 5 of our solution.
Back-substitution is the final step in piecing together the complete solution. Once we've determined the value of one variable, we can use it as a stepping stone to find the others. This iterative process allows us to systematically unravel the system of equations and arrive at the solution.
4. Ordered Triples
The solution to a system of three equations with three unknowns is represented as an ordered triple (x, y, z). This triple represents a point in three-dimensional space where all three equations intersect. We expressed our solution in this format in Step 6.
Representing solutions as ordered triples provides a clear and concise way to express the values of multiple variables. This notation is particularly useful in systems with three or more variables, where visualizing the solution can be challenging. The ordered triple represents a unique point that satisfies all the equations in the system.
5. Verification
It's always a good practice to verify your solution by substituting the values back into the original equations. This helps ensure that you haven't made any errors during the solving process. We verified our solution in the Verification section.
Verification is a crucial step in ensuring the accuracy of our solution. By plugging the values back into the original equations, we can confirm that they hold true. This process helps us catch any potential errors and build confidence in our answer.
Common Mistakes to Avoid
Solving systems of equations can be tricky, and it's easy to make mistakes along the way. Here are some common pitfalls to watch out for:
- Sign Errors: Be careful when dealing with negative signs. A small mistake in sign can lead to an incorrect solution.
- Incorrect Substitution: Make sure you substitute the expression for the correct variable into the correct equation.
- Arithmetic Errors: Double-check your arithmetic calculations, especially when adding, subtracting, multiplying, or dividing.
- Forgetting to Back-Substitute: Don't forget to back-substitute to find the values of all the variables.
- Not Verifying the Solution: Always verify your solution by substituting the values back into the original equations.
By being aware of these common mistakes, you can significantly improve your accuracy and problem-solving skills.
Conclusion
Solving systems of equations is a fundamental skill in mathematics with applications in various fields. By understanding the concepts and strategies outlined in this article, you can confidently tackle these problems. Remember to practice regularly and pay attention to detail to avoid common mistakes. With dedication and a systematic approach, you'll master the art of solving systems of equations.
We've walked through the process of solving a specific system of equations, highlighting the key steps and concepts involved. From isolating variables to back-substitution and verification, each step plays a crucial role in arriving at the correct solution. By mastering these techniques, you'll be well-equipped to solve a wide range of algebraic problems.
This step-by-step guide provides a solid foundation for understanding and solving systems of equations. Whether you're a student learning algebra or simply looking to refresh your math skills, the principles and strategies discussed here will prove invaluable. Remember to practice regularly and apply these techniques to different types of problems to solidify your understanding and enhance your problem-solving abilities. The more you practice, the more confident and proficient you'll become in solving systems of equations.
By consistently applying these strategies and paying attention to detail, you'll develop a strong foundation in solving systems of equations and excel in your mathematical endeavors. Keep practicing, and you'll find that even the most complex problems become manageable with the right approach and understanding.
