Solving The System Of Equations X + Y + Z = -2, 2x + 5y + 2z = -7, -x + 7y - 3z = -10 A Step-by-Step Guide

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\begin{aligned}
x+y+z & =-2 \\
2 x+5 y+2 z & =-7 \\
-x+7 y-3 z & =-10
\end{aligned}

To solve the given system of equations, we will use the method of elimination or substitution. Here's a step-by-step solution:

Step 1: Label the Equations

Let's label the equations for easy reference:

Equation 1: x + y + z = -2 Equation 2: 2x + 5y + 2z = -7 Equation 3: -x + 7y - 3z = -10

Step 2: Eliminate x from Equations 1 and 2

To eliminate x, we'll multiply Equation 1 by -2 and add it to Equation 2:

-2 * (x + y + z) = -2 * -2
-2x - 2y - 2z = 4

Now, add this modified equation to Equation 2:

(-2x - 2y - 2z) + (2x + 5y + 2z) = 4 + (-7) 3y = -3

Divide by 3 to solve for y:

y = -1

Step 3: Eliminate x from Equations 1 and 3

To eliminate x again, we'll add Equation 1 to Equation 3 directly:

(x + y + z) + (-x + 7y - 3z) = -2 + (-10) 8y - 2z = -12

Step 4: Substitute y = -1 into the Resulting Equation

Substitute y = -1 into the equation 8y - 2z = -12:

8(-1) - 2z = -12 -8 - 2z = -12

Add 8 to both sides:

-2z = -4

Divide by -2 to solve for z:

z = 2

Step 5: Substitute y and z into Equation 1

Substitute y = -1 and z = 2 into Equation 1:

x + (-1) + 2 = -2 x + 1 = -2

Subtract 1 from both sides to solve for x:

x = -3

Step 6: Verify the Solution

Let's verify the solution x = -3, y = -1, and z = 2 in all three original equations:

Equation 1: -3 + (-1) + 2 = -2 (Correct) Equation 2: 2(-3) + 5(-1) + 2(2) = -6 - 5 + 4 = -7 (Correct) Equation 3: -(-3) + 7(-1) - 3(2) = 3 - 7 - 6 = -10 (Correct)

Step 7: State the Solution

The solution to the system of equations is:

x = -3, y = -1, z = 2

Conclusion

Therefore, the correct choice is:

A. The solution is x = -3, y = -1, and z = 2.

Solving systems of equations is a fundamental skill in mathematics, and this step-by-step guide demonstrates how to tackle such problems systematically. Understanding these methods is crucial for various applications in engineering, economics, and computer science. Let's delve deeper into the intricacies of solving systems of equations, focusing on the method of elimination and its variations, to enhance our problem-solving skills.

The Importance of Solving Systems of Equations

In mathematics, solving systems of equations is a cornerstone skill with applications spanning numerous fields. A system of equations is a set of two or more equations containing the same variables. The solution to such a system is a set of values for the variables that satisfy all equations simultaneously. Mastery of these techniques is essential for students and professionals alike, as it forms the bedrock for more advanced mathematical concepts and real-world applications. Whether you are a student mastering algebraic concepts or a professional working on complex models, understanding how to solve these systems is invaluable.

The Method of Elimination: A Detailed Overview

The method of elimination is a powerful technique for solving systems of equations. It involves manipulating the equations such that, when added or subtracted, one or more variables are eliminated. This process simplifies the system into a more manageable form, allowing us to solve for the remaining variables. The key steps in the elimination method include:

  1. Preparation: Examine the system of equations and identify which variable is easiest to eliminate. This often involves finding a common multiple of the coefficients of the variable you want to eliminate.
  2. Manipulation: Multiply one or both equations by a constant so that the coefficients of the variable to be eliminated are additive inverses (e.g., 2x and -2x). This step is critical for ensuring that when the equations are combined, the targeted variable cancels out.
  3. Elimination: Add the modified equations. This should eliminate one variable, resulting in a new equation with one fewer variable.
  4. Back-Substitution: Solve the resulting equation for the remaining variable. Once you have the value of one variable, substitute it back into one of the original equations (or a modified version) to solve for another variable. Repeat this process until you have found the values for all variables.
  5. Verification: Substitute the values of all variables back into the original equations to ensure they satisfy all equations in the system. This step is crucial for catching any errors made during the solving process.

Applying the Method to Our System

Let's revisit our original system of equations:

x + y + z = -2
2x + 5y + 2z = -7
-x + 7y - 3z = -10

To begin, we can eliminate x from the first two equations. Multiply the first equation by -2 to get -2x - 2y - 2z = 4. Adding this to the second equation (2x + 5y + 2z = -7) gives us 3y = -3, which simplifies to y = -1. This exemplifies the manipulation and elimination steps at work.

Next, we eliminate x from the first and third equations by simply adding them: (x + y + z) + (-x + 7y - 3z) = -2 + (-10), which simplifies to 8y - 2z = -12. Substituting y = -1 into this equation gives -8 - 2z = -12. Solving for z, we find z = 2. This shows the back-substitution process.

Finally, substituting y = -1 and z = 2 into the first equation, x + y + z = -2, gives x - 1 + 2 = -2, which simplifies to x = -3. The solution (x = -3, y = -1, z = 2) demonstrates the successful application of the method of elimination.

Common Pitfalls and How to Avoid Them

Solving systems of equations can be tricky, and several common mistakes can lead to incorrect solutions. Here are some pitfalls to watch out for and strategies to avoid them:

  • Arithmetic Errors: Simple arithmetic mistakes can derail the entire solution process. Double-check each calculation, especially when dealing with negative numbers and fractions. Using a calculator or writing out each step can help reduce errors.
  • Incorrect Manipulation: Multiplying or dividing equations incorrectly can change the system, leading to a wrong solution. Ensure that you multiply or divide every term in the equation by the same constant. For instance, when we multiplied the first equation by -2, we applied it to every term: -2(x + y + z) = -2(-2) which resulted in -2x - 2y - 2z = 4.
  • Sign Errors: Sign errors are a common source of mistakes, particularly when adding or subtracting equations. Pay close attention to the signs of each term and use parentheses to avoid confusion.
  • Forgetting to Verify: Always verify your solution by substituting the values back into the original equations. This step will catch most arithmetic and manipulation errors.
  • Misinterpreting Solutions: In some cases, systems may have no solution or infinitely many solutions. Understanding how to identify these cases is crucial. For instance, if you end up with a contradiction (e.g., 0 = 1), the system has no solution. If you end up with an identity (e.g., 0 = 0), the system has infinitely many solutions.

Variations and Advanced Techniques

While the method of elimination is powerful, there are variations and advanced techniques that can be used to solve more complex systems of equations. Some of these include:

  • Substitution Method: Another fundamental technique involves solving one equation for one variable and substituting that expression into the other equations. This method is particularly useful when one equation is easily solved for one variable.
  • Matrix Methods: For larger systems of equations, matrix methods such as Gaussian elimination and matrix inversion provide a more systematic approach. These methods are often used in computer programs and numerical analysis.
  • Cramer's Rule: This method uses determinants to solve systems of linear equations. While it can be efficient for small systems, it becomes computationally intensive for larger systems.

Real-World Applications

Solving systems of equations is not just an academic exercise; it has numerous real-world applications. Some examples include:

  • Engineering: Engineers use systems of equations to design structures, analyze circuits, and model fluid flow.
  • Economics: Economists use systems of equations to model supply and demand, analyze market equilibrium, and forecast economic trends.
  • Computer Science: Computer scientists use systems of equations in various applications, including computer graphics, optimization problems, and network analysis.
  • Physics: Physicists use systems of equations to model physical systems, such as motion, electricity, and magnetism.
  • Chemistry: Chemists use systems of equations to balance chemical reactions and solve stoichiometry problems.

In conclusion, mastering the art of solving systems of equations is an invaluable skill. The method of elimination, with its systematic approach, provides a robust technique for tackling these problems. By understanding the nuances of the method, avoiding common pitfalls, and exploring advanced techniques, you can enhance your mathematical prowess and apply these skills to a wide range of real-world scenarios. Whether you are balancing chemical equations, designing structural components, or analyzing economic models, the ability to solve systems of equations will prove to be an indispensable asset.