Solving A System Of Equations: A Step-by-Step Guide

Hey guys! Ever find yourself staring blankly at a system of equations, wondering how to even begin solving it? Don't worry; we've all been there. Systems of equations might seem intimidating at first, but with a systematic approach, they become manageable. In this article, we're going to break down a specific system of equations and solve it step-by-step. So, buckle up, and let's dive in!

The System of Equations

Here’s the system we're going to tackle:

\begin{cases}
x - y - 2z = -2 \\
-x - y + z = 5 \\
3x + 2y - z = -1
\end{cases}

This system consists of three equations with three variables: x, y, and z. Our goal is to find the values of these variables that satisfy all three equations simultaneously. There are several methods to solve such systems, including substitution, elimination, and matrix methods. For this example, we'll use the elimination method, which is often straightforward and efficient.

Step 1: Elimination of x from Equations 2 and 3 using Equation 1

The elimination method involves adding or subtracting multiples of equations to eliminate one variable at a time. Let's start by eliminating x from the second and third equations using the first equation as a reference.

Eliminating x from Equation 2

To eliminate x from the second equation (-x - y + z = 5), we can simply add the first equation (x - y - 2z = -2) to it:

(x - y - 2z) + (-x - y + z) = -2 + 5

This simplifies to:

-2y - z = 3

Let's call this new equation Equation 4:

Equation 4: -2y - z = 3

Eliminating x from Equation 3

Now, let's eliminate x from the third equation (3x + 2y - z = -1). To do this, we'll multiply the first equation by -3 and add it to the third equation:

-3(x - y - 2z) = -3(-2)

-3x + 3y + 6z = 6

Now, add this to the third equation:

(-3x + 3y + 6z) + (3x + 2y - z) = 6 + (-1)

This simplifies to:

5y + 5z = 5

Divide the entire equation by 5 to simplify it further:

y + z = 1

Let's call this new equation Equation 5:

Equation 5: y + z = 1

Step 2: Solving for y and z

Now we have two equations with two variables, y and z:

Equation 4: -2y - z = 3

Equation 5: y + z = 1

We can solve this system using either substitution or elimination. Let's use the elimination method again. We can multiply Equation 5 by 2 to eliminate y:

2(y + z) = 2(1)

2y + 2z = 2

Now, add this new equation to Equation 4:

(-2y - z) + (2y + 2z) = 3 + 2

This simplifies to:

z = 5

So, we've found that z = 5. Now we can substitute this value back into Equation 5 to find y:

y + 5 = 1

y = 1 - 5

y = -4

So, we have y = -4 and z = 5.

Step 3: Solving for x

Now that we have the values for y and z, we can substitute them back into any of the original equations to solve for x. Let's use the first equation:

x - y - 2z = -2

Substitute y = -4 and z = 5:

x - (-4) - 2(5) = -2

x + 4 - 10 = -2

x - 6 = -2

x = -2 + 6

x = 4

So, we have x = 4.

Step 4: Verification

To ensure our solution is correct, we should substitute the values of x, y, and z back into all three original equations to verify they hold true.

Equation 1: x - y - 2z = -2

4 - (-4) - 2(5) = -2

4 + 4 - 10 = -2

8 - 10 = -2

-2 = -2 (True)

Equation 2: -x - y + z = 5

-4 - (-4) + 5 = 5

-4 + 4 + 5 = 5

0 + 5 = 5

5 = 5 (True)

Equation 3: 3x + 2y - z = -1

3(4) + 2(-4) - 5 = -1

12 - 8 - 5 = -1

4 - 5 = -1

-1 = -1 (True)

Since the values x = 4, y = -4, and z = 5 satisfy all three equations, our solution is correct.

Final Answer

The solution to the system of equations is:

x = 4, y = -4, z = 5

Or, as an ordered triple: (4, -4, 5)

So there you have it! Solving systems of equations becomes much easier with a step-by-step approach. Remember to eliminate variables systematically, solve for the remaining variables, and always verify your solution. Keep practicing, and you'll become a pro in no time. Happy solving!

Additional Tips for Solving Systems of Equations

1. Choose the Best Method

Different systems of equations are better suited for different methods. While we used elimination in this example, substitution can be more efficient when one of the equations is already solved for one variable. Matrix methods, like Gaussian elimination or using the inverse of a matrix, are particularly useful for larger systems.

2. Look for Simplifications

Before diving into solving, take a moment to see if any of the equations can be simplified. Dividing an equation by a common factor, for example, can make the numbers smaller and easier to work with. Simplifying early can reduce the chances of making errors later on.

3. Stay Organized

Solving systems of equations can involve a lot of steps, so it’s crucial to stay organized. Label your equations clearly, and keep track of your work. Writing neatly and systematically will help you avoid mistakes and make it easier to review your work if you get stuck.

4. Practice, Practice, Practice

The more you practice solving systems of equations, the more comfortable you'll become with the different techniques. Start with simpler systems and gradually work your way up to more complex ones. There are plenty of resources available online and in textbooks to help you practice.

5. Use Technology

Don't be afraid to use technology to help you solve systems of equations. There are many online calculators and software packages that can solve systems of equations quickly and accurately. While it's important to understand the underlying concepts, using technology can save you time and help you check your work.

6. Check for Special Cases

Sometimes, systems of equations have special cases. For example, a system might have no solution (inconsistent) or infinitely many solutions (dependent). Recognizing these cases early can save you time and effort. If you end up with a contradiction (e.g., 0 = 1) while solving, the system is inconsistent. If you end up with an identity (e.g., 0 = 0), the system is dependent.

7. Understand the Geometry

For systems of two equations with two variables, you can think of each equation as representing a line on a graph. The solution to the system is the point where the lines intersect. If the lines are parallel, there is no solution. If the lines are the same, there are infinitely many solutions. Visualizing the geometry can give you a better understanding of what's going on.

By following these tips and practicing regularly, you'll be well on your way to mastering systems of equations. Keep up the great work, and don't hesitate to ask for help when you need it!