Solving Systems Of Equations A Step By Step Guide

by Sharif Sakr 50 views

Hey there, math enthusiasts! Ever stared at a system of equations and felt like you're trying to decipher an ancient scroll? Fear not! We're about to embark on a journey to unravel the mysteries of solving systems of equations, turning those intimidating symbols into a clear path to the solution. In this article, we'll dive deep into the process of solving a specific system, breaking down each step with clear explanations and helpful tips. So, grab your pencils, put on your thinking caps, and let's get started!

The Challenge: Our System of Equations

Before we jump into the solution, let's take a good look at the system we're going to tackle. This will be our mathematical Everest, and we're going to conquer it together! Here it is:

-5x - 5y - 5z = -60
-15x + 5y - 10z = -4
-35x + 5y - 25z = -52

Okay, I know what you might be thinking: “Whoa, that looks complicated!” But trust me, we're going to break it down into manageable pieces. Each equation is a piece of the puzzle, and by working through them strategically, we'll reveal the values of x, y, and z that make all three equations true.

Step 1: Simplifying the Equations (Making Life Easier)

The first thing we can do to make our lives easier is to simplify the equations. Notice that the first equation, -5x - 5y - 5z = -60, has a common factor of -5 in every term. We can divide the entire equation by -5 to get a much simpler equation:

(-5x / -5) + (-5y / -5) + (-5z / -5) = (-60 / -5)

This simplifies beautifully to:

x + y + z = 12

See? Much friendlier already! Let's call this Equation 1 (our new, improved version). Now our system looks like this:

x + y + z = 12 (Equation 1)
-15x + 5y - 10z = -4 (Equation 2)
-35x + 5y - 25z = -52 (Equation 3)

Simplifying equations is a crucial step in solving systems. It reduces the size of the numbers we're working with, which means fewer opportunities for calculation errors. It's like decluttering your workspace before starting a big project – it makes everything feel more manageable.

Step 2: Elimination – Our Secret Weapon

The next step involves a powerful technique called elimination. The idea behind elimination is to get rid of one variable at a time by adding or subtracting multiples of equations. This might sound a bit mysterious, but it's actually quite straightforward. Look closely at Equations 2 and 3; both have a +5y term. This is our golden opportunity!

To eliminate the 'y' variable, we can subtract Equation 3 from Equation 2. This will cancel out the 'y' terms, leaving us with an equation involving only 'x' and 'z'. Let’s do it:

(-15x + 5y - 10z) - (-35x + 5y - 25z) = -4 - (-52)

Carefully distribute the negative sign and combine like terms:

-15x + 35x + 5y - 5y - 10z + 25z = -4 + 52

This simplifies to:

20x + 15z = 48

Let's call this Equation 4. We've successfully eliminated 'y' and created a new equation with just 'x' and 'z'. Elimination is a key strategy in solving systems, and identifying opportunities to eliminate variables is a crucial skill.

Step 3: More Elimination – Targeting Another Variable

Now we need another equation with only 'x' and 'z' to solve for those variables. We can use Equation 1 (x + y + z = 12) and either Equation 2 or Equation 3 to eliminate 'y' again. Let's use Equation 1 and Equation 2 this time. To eliminate 'y', we need to multiply Equation 1 by -5 so that the 'y' term becomes -5y, which will cancel with the +5y in Equation 2.

Multiply Equation 1 by -5:

-5(x + y + z) = -5(12)

This gives us:

-5x - 5y - 5z = -60

Now, let's add this modified Equation 1 to Equation 2:

(-5x - 5y - 5z) + (-15x + 5y - 10z) = -60 + (-4)

Combine like terms:

-20x - 15z = -64

Let's call this Equation 5. Notice that Equations 4 and 5 both involve only 'x' and 'z'. We're getting closer!

Step 4: Solving the Two-Variable System

We now have a system of two equations with two variables (x and z):

20x + 15z = 48 (Equation 4)
-20x - 15z = -64 (Equation 5)

This looks almost too good to be true! If we add Equation 4 and Equation 5, both the 'x' and 'z' terms will cancel out:

(20x + 15z) + (-20x - 15z) = 48 + (-64)

This simplifies to:

0 = -16

Wait a minute... 0 = -16? That's not right! This is a contradiction, which means our system of equations has no solution. In other words, there are no values for x, y, and z that will satisfy all three original equations simultaneously.

Sometimes, in math (and in life!), you encounter situations where there's no straightforward answer. Recognizing a contradiction like this is just as important as finding a solution. It tells us something fundamental about the relationship between the equations in our system.

Step 5: Understanding the No Solution Outcome

So, what does it mean when a system has no solution? Geometrically, each linear equation in three variables represents a plane in 3D space. When we solve a system of three equations, we're looking for the point where all three planes intersect. If there's no solution, it means these planes don't all intersect at a single point. They might intersect in pairs, or they might be parallel, but they don't share a common intersection point.

In our case, the contradiction we encountered tells us that the equations are inconsistent. They describe a situation that is mathematically impossible. This is a valuable piece of information, and it's important to recognize it when it occurs.

Key Takeaways and Tips for System Solving

We may not have found a numerical solution in this case, but we've learned a lot about solving systems of equations. Here are some key takeaways and tips to keep in mind:

  • Simplify First: Always look for opportunities to simplify equations by dividing out common factors. This makes the numbers smaller and easier to work with.
  • Master Elimination: Elimination is a powerful technique for reducing the number of variables in a system. Look for opportunities to add or subtract multiples of equations to cancel out variables.
  • Stay Organized: Keep track of your equations and label them clearly. This will help you avoid confusion and make it easier to follow your steps.
  • Recognize Contradictions: If you encounter a contradiction (like 0 = -16), it means the system has no solution. Don't try to force a solution that doesn't exist.
  • Visualize the Geometry: Remember that systems of linear equations have a geometric interpretation. Thinking about lines, planes, and intersections can help you understand what's happening algebraically.

Solving systems of equations is a fundamental skill in mathematics and has applications in many fields, including engineering, physics, economics, and computer science. By understanding the techniques and strategies involved, you'll be well-equipped to tackle a wide range of problems.

Wrapping Up: The Beauty of the Process

Even though our system didn't have a solution, the process of working through it was incredibly valuable. We honed our skills in simplification, elimination, and recognizing contradictions. We learned that sometimes, the journey is just as important as the destination.

So, the next time you encounter a system of equations, remember these steps, embrace the challenge, and enjoy the process of cracking the code! Keep practicing, keep exploring, and keep pushing your mathematical boundaries. You've got this!