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

Hey guys! Let's dive into solving systems of equations. It might sound intimidating, but trust me, it's totally doable. We're going to break down a specific example step by step so you can tackle these problems like a pro. We will solve the system of equations for all three variables. The system is:

2x - 5y + 10z = -4
-2x + y - 8z = 0
2x + 6y + 9z = -2

Understanding Systems of Equations

Before we jump into the solution, let's quickly recap what a system of equations is. Basically, it's a set of two or more equations that involve the same variables. Our goal is to find the values for these variables that satisfy all equations simultaneously. In simpler terms, we want to find the x, y, and z values that make all three equations true at the same time. This involves using methods like elimination, substitution, or matrix operations to systematically reduce the equations until we isolate each variable. Mastering the art of solving systems of equations opens doors to tackling real-world problems across various fields, including engineering, economics, and computer science, where relationships between multiple variables need to be understood and quantified. It's like unlocking a superpower for problem-solving!

Why Solving Systems of Equations Matters

Solving systems of equations is crucial in various fields, and you might be wondering why we even bother learning this stuff. Well, the beauty of these systems lies in their ability to model real-world scenarios. Think about it: many situations involve multiple variables that are interconnected. For instance, in economics, you might have supply and demand equations that determine market equilibrium. In physics, you might have equations describing the motion of objects under different forces. By learning how to solve systems of equations, you're gaining a powerful tool to analyze and understand these complex relationships. It’s the cornerstone of mathematical modeling, allowing us to predict outcomes, optimize processes, and make informed decisions. The applications are virtually limitless, ranging from optimizing traffic flow in transportation networks to designing efficient circuits in electronics. The ability to translate real-world problems into a mathematical framework and solve them is a skill that will serve you well in countless scenarios.

Step 1: Elimination Method – Targeting 'x'

The elimination method is our first key technique. This method involves adding or subtracting equations to eliminate one variable, making the system simpler to solve. We’ll start by targeting the variable x because it has convenient coefficients in our system. Notice how the first and second equations have 2x and -2x, respectively. This makes it easy to eliminate x by simply adding these two equations together.

Adding Equations 1 and 2

Let's add the first and second equations:

(2x - 5y + 10z) + (-2x + y - 8z) = -4 + 0

This simplifies to:

-4y + 2z = -4

We'll call this new equation Equation 4.

Adding Equations to Eliminate x

Now, let's consider the first and third equations. Both have a 2x term. To eliminate x here, we need to subtract the first equation from the third equation:

(2x + 6y + 9z) - (2x - 5y + 10z) = -2 - (-4)

This simplifies to:

11y - z = 2

We'll call this Equation 5. By strategically adding and subtracting equations, we've successfully eliminated x and created a new system of two equations (Equations 4 and 5) with two variables (y and z). This simplification is a crucial step in solving the original system, making the problem more manageable and paving the way for finding the values of y and z. The art of choosing which equations to combine and how lies in identifying the coefficients that will conveniently cancel out a variable, guiding you closer to the solution.

Step 2: Solving for 'y' and 'z'

Now we have a simpler system of two equations with two variables:

-4y + 2z = -4  (Equation 4)
11y - z = 2    (Equation 5)

To solve for y and z, we can use the elimination method again. This time, let's eliminate z. To do this, multiply Equation 5 by 2:

Multiplying Equation 5 by 2

2 * (11y - z) = 2 * 2

This gives us:

22y - 2z = 4   (Equation 6)

Adding Equations 4 and 6

Now add Equation 4 and Equation 6:

(-4y + 2z) + (22y - 2z) = -4 + 4

This simplifies to:

18y = 0

Solving for 'y'

Divide both sides by 18:

y = 0

So, we've found that y = 0! Knowing the value of y is a significant step forward. Now we can substitute this value into one of our equations (Equation 4 or 5) to solve for z. Let's use Equation 5 because it looks simpler.

Substituting 'y' into Equation 5

11(0) - z = 2

This simplifies to:

-z = 2

Solving for 'z'

Multiply both sides by -1:

z = -2

Fantastic! We've found that z = -2. With the values of y and z in hand, we're one step closer to completing the puzzle. We can now use these values to find the value of x by substituting them into one of the original equations.

Step 3: Solving for 'x'

We now know that y = 0 and z = -2. To find x, we can substitute these values into any of the original three equations. Let's use the first equation because it's there and ready to go:

2x - 5y + 10z = -4

Substituting 'y' and 'z'

Plug in y = 0 and z = -2:

2x - 5(0) + 10(-2) = -4

This simplifies to:

2x - 0 - 20 = -4

Simplifying the Equation

2x - 20 = -4

Add 20 to both sides:

2x = 16

Solving for 'x'

Divide both sides by 2:

x = 8

Excellent! We've found that x = 8. Now we have all the pieces of the puzzle: x = 8, y = 0, and z = -2. But before we celebrate, it's crucial to verify our solution to ensure we haven't made any errors along the way. This step provides confidence in our answer and validates the accuracy of our calculations.

Step 4: Verifying the Solution

To make sure our solution is correct, we need to substitute the values x = 8, y = 0, and z = -2 back into all three original equations. If the equations hold true, then our solution is correct. Let's start with the first equation:

Verifying in Equation 1

2x - 5y + 10z = -4

Substitute the values:

2(8) - 5(0) + 10(-2) = -4

Simplify:

16 - 0 - 20 = -4

-4 = -4

The first equation holds true! That's a good start. Now let's check the second equation.

Verifying in Equation 2

-2x + y - 8z = 0

Substitute the values:

-2(8) + 0 - 8(-2) = 0

Simplify:

-16 + 0 + 16 = 0

0 = 0

The second equation also holds true! We're on a roll. Finally, let's verify the solution in the third equation.

Verifying in Equation 3

2x + 6y + 9z = -2

Substitute the values:

2(8) + 6(0) + 9(-2) = -2

Simplify:

16 + 0 - 18 = -2

-2 = -2

The third equation holds true as well! This confirms that our solution x = 8, y = 0, and z = -2 is correct. Verifying the solution is not just a formality; it’s a crucial step that ensures the accuracy of our work and provides peace of mind. It's like a final checkmark that gives us confidence in our answer and the process we used to obtain it.

Solution

The solution to the system of equations is:

x = 8
y = 0
z = -2

We can write this as an ordered triple: (8, 0, -2).

Conclusion

So, there you have it! We've successfully solved the system of equations using the elimination method. Remember, solving systems of equations is a valuable skill that can be applied in many areas. Keep practicing, and you'll become a pro in no time! The key takeaway here is the systematic approach we adopted. By breaking down the problem into manageable steps – eliminating variables one at a time, solving for the remaining variables, and finally verifying the solution – we navigated through the complexity with clarity and precision. The process may seem lengthy at first, but with practice, it becomes second nature. Each step builds upon the previous one, creating a logical flow that leads to the solution. So, the next time you encounter a system of equations, remember the power of a structured approach and the satisfaction of finding the solution. You've got this! Keep exploring, keep learning, and keep solving!