When faced with a system of equations, the goal is to find the values of the variables that satisfy all equations simultaneously. In this article, we will explore a detailed method to solve the given system of equations:
\begin{aligned}
x + y + z &= 9 \\
2x + 4y + 2z &= 26 \\
-x + 7y - 3z &= 13
\end{aligned}
This system consists of three linear equations with three unknowns (x, y, and z). We will use a combination of techniques, including elimination and substitution, to arrive at the solution. Solving systems of equations is a fundamental skill in mathematics, with applications spanning various fields, including engineering, economics, and computer science. Understanding the methodologies involved not only enhances problem-solving capabilities but also provides a strong foundation for more advanced mathematical concepts. The ability to systematically approach and solve these systems is invaluable for both academic pursuits and practical applications.
Step 1: Simplify the Equations
Before diving into the elimination method, it's beneficial to simplify the equations if possible. Observe the second equation:
2x + 4y + 2z = 26
Each term is divisible by 2. Dividing the entire equation by 2 simplifies it to:
x + 2y + z = 13
This simplification makes the equation easier to work with in subsequent steps. Simplifying equations at the outset can significantly reduce the complexity of the calculations involved. By reducing coefficients and constants, the likelihood of making errors is diminished, and the overall process becomes more manageable. This step exemplifies an important strategy in problem-solving: always look for ways to make the problem simpler before tackling it head-on. By simplifying, we not only reduce computational burden but also gain a clearer perspective on the structure of the problem, making it easier to identify the most efficient solution path.
Our system now looks like this:
\begin{aligned}
x + y + z &= 9 &\text{(1)} \\
x + 2y + z &= 13 &\text{(2)} \\
-x + 7y - 3z &= 13 &\text{(3)}
\end{aligned}
Step 2: Eliminate 'x' from Equations (2) and (3)
The next step involves eliminating one variable to reduce the system to two equations with two unknowns. We'll start by eliminating 'x'.
To eliminate 'x' from Equation (2), subtract Equation (1) from Equation (2):
(x + 2y + z) - (x + y + z) = 13 - 9
This simplifies to:
y = 4
Now, let’s eliminate 'x' from Equation (3). Add Equation (1) to Equation (3):
(-x + 7y - 3z) + (x + y + z) = 13 + 9
This simplifies to:
8y - 2z = 22
Eliminating variables is a crucial technique in solving systems of equations. It systematically reduces the complexity of the problem by decreasing the number of unknowns. By strategically adding or subtracting multiples of equations, we can cancel out specific variables, making it easier to isolate and solve for the remaining variables. This process not only streamlines the solution but also provides a clear and organized approach to tackling complex systems. The ability to eliminate variables efficiently is a testament to a strong understanding of algebraic manipulation and is an essential skill for anyone working with multi-variable equations.
We now have a simpler system:
\begin{aligned}
y &= 4 &\text{(4)} \\
8y - 2z &= 22 &\text{(5)}
\end{aligned}
Step 3: Solve for 'z'
We already have the value of 'y' from Equation (4). Substitute y = 4 into Equation (5):
8(4) - 2z = 22
32 - 2z = 22
Subtract 32 from both sides:
-2z = -10
Divide by -2:
z = 5
Solving for variables involves isolating the unknown quantity by performing inverse operations. In this step, we substituted the known value of 'y' into the reduced equation to solve for 'z'. This highlights the power of substitution as a method for finding the values of unknowns in a system. By carefully applying algebraic manipulations, we can peel away the layers of the equation to reveal the value of the desired variable. The methodical approach ensures accuracy and efficiency in the solution process, and it is a testament to the underlying principles of algebraic problem-solving. The ability to solve for variables is a core skill in mathematics and is essential for tackling a wide range of problems in various fields.
Step 4: Solve for 'x'
Now that we have the values of y and z, we can substitute them into Equation (1) to find 'x':
x + y + z = 9
x + 4 + 5 = 9
x + 9 = 9
Subtract 9 from both sides:
x = 0
Finding the value of 'x' completes the solution process by utilizing the previously determined values of 'y' and 'z'. Substituting these values into one of the original equations allows us to isolate 'x' and solve for its value. This step demonstrates the interconnectedness of variables within a system of equations and how solving for one variable can pave the way for solving for others. The substitution method is a powerful tool for navigating these relationships and arriving at a complete solution. The methodical approach ensures accuracy and consistency, and it underscores the importance of a systematic strategy in solving complex mathematical problems.
Step 5: Verify the Solution
To ensure our solution is correct, we substitute the values of x, y, and z into all three original equations:
Equation (1):
0 + 4 + 5 = 9 (Correct)
Equation (2):
2(0) + 4(4) + 2(5) = 0 + 16 + 10 = 26 (Correct)
Equation (3):
-0 + 7(4) - 3(5) = 0 + 28 - 15 = 13 (Correct)
Since the solution satisfies all three equations, it is correct.
Verifying the solution is a critical step in the problem-solving process. By substituting the obtained values back into the original equations, we ensure that the solution is consistent and accurate. This process acts as a safeguard against errors made during the solution process and provides confidence in the final result. Verification not only confirms the correctness of the solution but also reinforces the understanding of the relationships between variables in the system. This step is essential for maintaining mathematical rigor and is a hallmark of a thorough and meticulous problem-solving approach.
Final Answer
The solution to the system of equations is:
x = 0, y = 4, z = 5
In this comprehensive guide, we have walked through the process of solving a system of three linear equations with three unknowns. The techniques used, such as simplification, elimination, substitution, and verification, are fundamental to solving various mathematical problems. Mastering systems of equations is not just about finding the right answers; it's about developing a methodical and logical approach to problem-solving.
The ability to solve systems of equations is a cornerstone of mathematical proficiency and has widespread applications across diverse fields. From engineering and physics to economics and computer science, the skills developed in solving these systems are invaluable. The process of simplifying equations, eliminating variables, and verifying solutions cultivates critical thinking and analytical skills that are essential for success in both academic and professional endeavors. Moreover, the understanding gained from solving systems of equations lays the groundwork for more advanced mathematical concepts, making it a crucial area of study for anyone pursuing a career in a STEM field.
By understanding and applying these methods, you can confidently tackle similar problems and enhance your mathematical skills. Remember to practice regularly and apply these techniques to various problems to solidify your understanding. Consistent practice is key to mastering any mathematical concept, and systems of equations are no exception. By working through a variety of examples, you will not only become more proficient in applying the techniques but also develop a deeper intuition for the underlying principles. This hands-on experience will build your confidence and make you a more effective problem solver. Moreover, the ability to adapt and apply these methods to real-world scenarios will further enhance your understanding and demonstrate the practical relevance of mathematical skills.