Solving For X: System Of Equations Example

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Hey guys! Today, we're diving into a classic math problem: solving for the value of 'x' in a system of three linear equations. This is a fundamental skill in algebra and can come in handy in various real-world scenarios, from balancing chemical equations to modeling economic systems. So, let's break down the problem step-by-step and make sure we understand the core concepts involved.

The Problem

We're given the following system of equations:

  1. x + y + z = 1
  2. 2x - y - z = -5
  3. 2x - 2y - z = 7

Our mission, should we choose to accept it (and we do!), is to find the value of 'x'. There are several ways we can tackle this, but we'll focus on a method called elimination, which is super efficient for this type of problem. Elimination involves strategically adding or subtracting equations to get rid of variables. The main goal is to reduce the system to a single equation with only one unknown.

The Elimination Method: Step-by-Step

Step 1: Eliminate 'y' and 'z' from two equations

Let's start by eliminating 'y' and 'z' from equations 1 and 2. Notice that the 'y' and 'z' terms have opposite signs in these equations, which is perfect for elimination! If we simply add equations 1 and 2 together, the 'y' and 'z' terms will cancel out:

(x + y + z) + (2x - y - z) = 1 + (-5)

This simplifies to:

3x = -4

Now we have a simple equation with just 'x'! But hold on, we're not done yet. We need to use all three equations to ensure we have a consistent solution. So, let's move on to the next step.

Step 2: Eliminate 'y' and 'z' from a different pair of equations

This time, let's use equations 1 and 3. However, we can't directly add them because the coefficients of 'y' don't match. To eliminate 'y', we need to multiply equation 1 by 2:

2 * (x + y + z) = 2 * 1

This gives us:

  1. 2x + 2y + 2z = 2

Now we can add equation 3 to this new equation (equation 4):

(2x + 2y + 2z) + (2x - 2y - z) = 2 + 7

This simplifies to:

4x + z = 9

We now have another equation, but it still has two variables ('x' and 'z'). This is where things get a little trickier, but don't worry, we've got this!

Step 3: Eliminate 'z' to solve for 'x'

We need to find a way to eliminate 'z' from the equation we just derived (4x + z = 9) and the equation we got in step 1 (3x = -4). But wait! We already have an equation with just 'x' (3x = -4). This means we can directly solve for 'x'!

Solving for x

From the equation 3x = -4, we can solve for 'x' by dividing both sides by 3:

x = -4/3

Eureka! We've found the value of 'x'.

Final Answer

The value of x in the given system of equations is -4/3. So the answer is B. -4/3.

Why This Works: A Deeper Dive

The elimination method works because we're essentially manipulating the equations in a way that preserves their equality. When we add or subtract equations, we're adding or subtracting the same quantity from both sides of the equation, which doesn't change the solution. The strategic part is choosing which equations to combine and how to combine them to eliminate variables. This process systematically reduces the complexity of the system until we can isolate the variable we're interested in.

Think of it like a puzzle where each equation is a piece of information. By carefully combining these pieces, we can reveal the hidden solution.

Alternative Methods (Briefly) and Why Elimination is Often Preferred

While elimination is a powerful technique, there are other ways to solve systems of equations. Two common alternatives are:

  1. Substitution: This involves solving one equation for one variable and then substituting that expression into another equation. This method can be useful when one of the equations is already solved for a variable or when it's easy to isolate a variable.
  2. Matrices: Systems of equations can be represented as matrices, and techniques from linear algebra (like Gaussian elimination or finding the inverse of a matrix) can be used to solve for the variables. This method is particularly useful for larger systems of equations with many variables.

So, why did we choose elimination here? In this case, elimination was a natural choice because the 'y' and 'z' terms had opposite signs in equations 1 and 2, making them easy to eliminate. This allowed us to quickly reduce the system to a simpler form.

Common Mistakes to Avoid

When solving systems of equations, it's easy to make small errors that can lead to incorrect answers. Here are a few common pitfalls to watch out for:

  • Arithmetic errors: Double-check your addition, subtraction, multiplication, and division. Even a small mistake can throw off the entire solution.
  • Sign errors: Pay close attention to the signs of the terms when adding or subtracting equations. A misplaced negative sign can be a disaster.
  • Incorrectly distributing: When multiplying an equation by a constant, make sure to distribute the constant to every term in the equation.
  • Not checking your answer: After you've found a solution, plug it back into the original equations to make sure it works. This is a crucial step for catching errors.

Practice Makes Perfect!

The best way to master solving systems of equations is to practice! Work through plenty of examples, and don't be afraid to make mistakes. Each mistake is a learning opportunity. The more you practice, the more comfortable and confident you'll become.

Real-World Applications

You might be wondering, "When am I ever going to use this in real life?" Well, systems of equations pop up in all sorts of unexpected places!

  • Engineering: Engineers use systems of equations to analyze circuits, design structures, and model fluid flow.
  • Economics: Economists use systems of equations to model supply and demand, analyze market equilibrium, and forecast economic trends.
  • Computer graphics: Systems of equations are used to create realistic images and animations.
  • Chemistry: Balancing chemical equations involves solving a system of equations.
  • Even cooking! You might use a system of equations to adjust a recipe for a different number of servings.

The ability to solve systems of equations is a valuable skill that can open doors to many different fields.

Let's Summarize: Key Takeaways

  • The elimination method is a powerful technique for solving systems of equations.
  • The goal of elimination is to get rid of variables by strategically adding or subtracting equations.
  • Careful arithmetic and attention to signs are crucial for avoiding errors.
  • Practice is the key to mastering this skill.
  • Systems of equations have many real-world applications.

Conclusion

So there you have it, guys! We've successfully solved for 'x' in a system of equations using the elimination method. Remember, the key is to break down the problem into smaller, manageable steps and to be meticulous in your calculations. Keep practicing, and you'll become a system-of-equations-solving pro in no time!