Solving Systems Of Equations A Step-by-Step Guide

by ADMIN 50 views

Hey guys! Ever felt like you're juggling multiple balls in the air, trying to solve a puzzle with too many pieces? That's how systems of equations can feel sometimes. But don't worry, we're going to break it down and make it super easy to understand. Today, we're tackling a specific system, but the techniques we'll use can be applied to tons of different problems. So, let's dive in!

The System We're Solving

First, let's get the problem on the table. We're going to solve the following system of equations:

3x + 3y + z = -21
x - 3y + 2z = 0
8x - 2y + 3z = -27

This might look a little intimidating at first glance, but trust me, it's totally manageable. We have three equations and three unknowns (x, y, and z), which means we can find a unique solution. Our goal is to find the values of x, y, and z that satisfy all three equations simultaneously. There are a few methods we can use to tackle this, and we're going to focus on the elimination method here.

Why the Elimination Method?

The elimination method is a powerful technique for solving systems of equations. The basic idea is to manipulate the equations in a way that allows us to eliminate one variable at a time. We do this by multiplying equations by constants and then adding or subtracting them. This will give us a new equation with fewer variables, which is much easier to solve. It’s like strategically taking pieces off a puzzle to reveal the bigger picture. For systems with three or more variables, elimination is often the most efficient approach. It's systematic and helps avoid getting bogged down in complex algebra. Plus, it's a great way to build your problem-solving muscles!

Step-by-Step Solution Using Elimination

Okay, let's get our hands dirty and walk through the solution step-by-step. I'll break it down into manageable chunks so it's crystal clear.

Step 1: Eliminate 'y' from Equations 1 and 2

Notice that the 'y' terms in the first two equations have coefficients of +3 and -3. This is perfect for elimination! If we simply add these two equations together, the 'y' terms will cancel out.

Equation 1: 3x + 3y + z = -21 Equation 2: x - 3y + 2z = 0

Adding them gives us:

(3x + 3y + z) + (x - 3y + 2z) = -21 + 0 4x + 3z = -21

Let's call this new equation Equation 4. We've successfully eliminated 'y' and now have an equation with just 'x' and 'z'.

Step 2: Eliminate 'y' Again, This Time from Equations 1 and 3

To eliminate 'y' again, we need to make the coefficients of 'y' in Equations 1 and 3 opposites of each other. Equation 1 has a +3y term, and Equation 3 has a -2y term. The least common multiple of 3 and 2 is 6, so we'll multiply Equation 1 by 2 and Equation 3 by 3 to get +6y and -6y terms, respectively.

Multiply Equation 1 by 2: 2 * (3x + 3y + z) = 2 * -21 => 6x + 6y + 2z = -42 Multiply Equation 3 by 3: 3 * (8x - 2y + 3z) = 3 * -27 => 24x - 6y + 9z = -81

Now, add these two modified equations:

(6x + 6y + 2z) + (24x - 6y + 9z) = -42 + (-81) 30x + 11z = -123

We'll call this Equation 5. We've eliminated 'y' again and have another equation with just 'x' and 'z'.

Step 3: Solve the System of Two Equations (Equations 4 and 5)

Now we have two equations with two unknowns:

Equation 4: 4x + 3z = -21 Equation 5: 30x + 11z = -123

We can use the elimination method again! Let's eliminate 'z'. To do this, we'll multiply Equation 4 by 11 and Equation 5 by -3.

Multiply Equation 4 by 11: 11 * (4x + 3z) = 11 * -21 => 44x + 33z = -231 Multiply Equation 5 by -3: -3 * (30x + 11z) = -3 * -123 => -90x - 33z = 369

Add these two equations:

(44x + 33z) + (-90x - 33z) = -231 + 369 -46x = 138

Now, solve for 'x':

x = 138 / -46 x = -3

We've found our first variable! Now we can substitute this value back into either Equation 4 or 5 to solve for 'z'. Let's use Equation 4:

4 * (-3) + 3z = -21 -12 + 3z = -21 3z = -9 z = -3

We've got 'z' too!

Step 4: Substitute 'x' and 'z' Back into One of the Original Equations to Solve for 'y'

Now that we know 'x' and 'z', we can plug them back into any of the original three equations to solve for 'y'. Let's use Equation 2, since it looks the simplest:

x - 3y + 2z = 0 (-3) - 3y + 2 * (-3) = 0 -3 - 3y - 6 = 0 -3y = 9 y = -3

We've found 'y'!

The Solution!

We did it! We've successfully solved the system of equations. The solution is:

  • x = -3
  • y = -3
  • z = -3

So, the solution can be written as an ordered triple: (-3, -3, -3). That wasn't so bad, was it?

Checking Our Answer

It's always a good idea to check our answer to make sure we didn't make any mistakes along the way. To do this, we'll substitute our values for x, y, and z back into all three original equations and see if they hold true.

Equation 1:

3 * (-3) + 3 * (-3) + (-3) = -21 -9 - 9 - 3 = -21 -21 = -21 (Correct!)

Equation 2:

(-3) - 3 * (-3) + 2 * (-3) = 0 -3 + 9 - 6 = 0 0 = 0 (Correct!)

Equation 3:

8 * (-3) - 2 * (-3) + 3 * (-3) = -27 -24 + 6 - 9 = -27 -27 = -27 (Correct!)

Our solution checks out in all three equations, so we can be confident that (-3, -3, -3) is the correct solution.

Common Mistakes to Avoid

When solving systems of equations, it's easy to make small errors that can throw off your entire solution. Here are a few common pitfalls to watch out for:

  • Arithmetic Errors: Be extra careful with your arithmetic, especially when dealing with negative numbers. A simple sign error can lead to a completely wrong answer. Double-check your calculations, and don't be afraid to use a calculator if needed.
  • Forgetting to Distribute: When multiplying an equation by a constant, make sure you distribute the constant to every term in the equation. It's easy to forget a term, especially if you're working quickly.
  • Incorrectly Combining Equations: Make sure you're adding or subtracting equations correctly. Pay attention to the signs of the terms. If you're subtracting an equation, remember to distribute the negative sign to every term in the equation.
  • Not Checking Your Solution: Always, always check your solution by substituting it back into the original equations. This is the best way to catch errors and ensure that your answer is correct.

By being mindful of these common mistakes, you can increase your accuracy and solve systems of equations with confidence.

Different Methods for Solving Systems of Equations

While we focused on the elimination method in this article, it's not the only tool in the toolbox. There are other methods you can use to solve systems of equations, each with its own strengths and weaknesses. Here are a few popular alternatives:

  • Substitution Method: In the substitution method, you solve one equation for one variable and then substitute that expression into another equation. This reduces the system to a single equation with one variable, which you can then solve. Substitution is often a good choice when one of the equations is already solved for a variable or can be easily solved.
  • Graphical Method: For systems of two equations with two variables, you can graph the equations on a coordinate plane. The solution to the system is the point where the graphs intersect. The graphical method is a great visual way to understand the solution, but it may not be practical for systems with more than two variables or when the solutions are not integers.
  • Matrix Methods: For larger systems of equations, matrix methods like Gaussian elimination or matrix inversion can be very efficient. These methods involve representing the system as a matrix and then performing operations on the matrix to solve for the variables. Matrix methods are commonly used in computer programs and applications that solve systems of equations.

The best method to use depends on the specific system of equations you're dealing with. Practice with different methods to develop your problem-solving skills and learn to choose the most efficient approach.

Real-World Applications of Systems of Equations

Systems of equations aren't just abstract math problems; they pop up in tons of real-world situations. Think about it: many situations involve multiple variables and relationships between them. That's where systems of equations come to the rescue! Here are just a few examples:

  • Mixing Problems: Imagine you're a chemist mixing solutions with different concentrations of a substance. You might need to solve a system of equations to figure out how much of each solution to mix to get the desired concentration.
  • Distance, Rate, and Time Problems: If you have two objects moving at different speeds and you know the total distance they travel and the total time, you can use a system of equations to find their individual speeds.
  • Supply and Demand in Economics: Economists use systems of equations to model the relationship between the supply and demand of goods and services. The solution to the system represents the equilibrium price and quantity.
  • Circuit Analysis in Electrical Engineering: Electrical engineers use systems of equations to analyze circuits and determine the currents and voltages in different parts of the circuit.
  • Curve Fitting in Statistics: Statisticians use systems of equations to find the equation of a curve that best fits a set of data points.

These are just a few examples, but the possibilities are endless. Systems of equations are a fundamental tool in many fields, and mastering them will open doors to a wide range of problem-solving opportunities.

Conclusion

So, we've successfully solved a system of three equations with three unknowns using the elimination method. We broke down each step, checked our answer, and even talked about common mistakes to avoid. Remember, practice makes perfect! The more you work with systems of equations, the more comfortable and confident you'll become. Keep practicing, and you'll be solving systems of equations like a pro in no time! And remember, math can be fun – especially when you conquer a tough problem. Keep up the great work, guys!