Solving System Of Equations Find Values Of X And Y

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Hey guys! Let's dive into how to solve a system of equations. It might sound intimidating, but trust me, it's totally manageable. We're going to break it down step by step so you can nail it every time. This article focuses on finding the values of x and y that satisfy a given system of equations. We will walk through the process step-by-step, ensuring you understand each stage and can apply the same techniques to similar problems.

Understanding the Basics of Systems of Equations

First, let's get clear on what a system of equations actually is. Think of it as a set of two or more equations that have the same variables. Our goal? To find the values for those variables that make all the equations true at the same time. It's like finding the perfect combo that unlocks all the equations.

In this case, we have two equations:

  1. ½x + y = 5
  2. x - y = 4

We need to find the values of x and y that work for both of these equations. There are a couple of common methods to tackle this: substitution and elimination. Let's explore them!

Methods for Solving Systems of Equations

There are primarily two methods we can use to solve systems of equations: substitution and elimination. Both methods aim to simplify the equations so we can solve for one variable at a time. Understanding these methods is crucial for mastering algebra and solving real-world problems involving multiple variables.

1. The Substitution Method

The substitution method is all about isolating one variable in one equation and then plugging that expression into the other equation. This way, we turn a two-variable problem into a single-variable problem, which is way easier to solve. Think of it like swapping out one thing for another to make the whole puzzle simpler.

Here’s how it works:

  1. Solve one equation for one variable: Look for an equation where it's easy to isolate either x or y. In our example, the second equation (x - y = 4) looks promising. We can easily solve for x by adding y to both sides: x = y + 4.
  2. Substitute: Now, take that expression (y + 4) and plug it in for x in the other equation (½x + y = 5). This gives us ½(y + 4) + y = 5.
  3. Solve for the remaining variable: Simplify and solve the new equation for y. We’ll walk through the steps in detail shortly.
  4. Back-substitute: Once you've found the value of y, plug it back into either of the original equations (or the rearranged one, like x = y + 4) to find x.

2. The Elimination Method

The elimination method, also known as the addition method, is super handy when you can add or subtract the equations in a way that cancels out one of the variables. It’s like strategically eliminating a piece of the puzzle to reveal the solution. This method is particularly effective when the coefficients of one variable are the same or easily made the same.

Here’s the breakdown:

  1. Line up the equations: Make sure the x and y terms are aligned in columns.
  2. Multiply (if necessary): If the coefficients of either x or y are not the same (or opposites), multiply one or both equations by a constant so that they are. The goal is to have either the x coefficients or the y coefficients be the same number but with opposite signs (e.g., 2 and -2).
  3. Add or subtract the equations: If the coefficients have opposite signs, add the equations. If they have the same sign, subtract the equations. This will eliminate one variable.
  4. Solve for the remaining variable: You’ll now have a single equation with one variable. Solve it!
  5. Back-substitute: Plug the value you found back into either of the original equations to solve for the other variable.

Step-by-Step Solution Using the Elimination Method

Let's tackle our system of equations using the elimination method. Remember, we have:

  1. ½x + y = 5
  2. x - y = 4

1. Line Up the Equations

Good news! Our equations are already nicely lined up with x terms over x terms and y terms over y terms.

2. Multiply (If Necessary)

Notice that the coefficients of y are already opposites (+1 and -1). This means we can skip this step – lucky us!

3. Add the Equations

Now, let's add the two equations together:

(½x + y) + (x - y) = 5 + 4

This simplifies to:

½x + x = 9

  1. 5x = 9

4. Solve for the Remaining Variable (x)

To isolate x, we combine the x terms and then divide both sides by 1.5:

  1. 5x = 9

x = 9 / 1.5

x = 6

Awesome! We’ve found that x = 6.

5. Back-Substitute to Find y

Now that we know x, we can plug it back into either of the original equations to solve for y. Let’s use the second equation, x - y = 4, because it looks a bit simpler:

6 - y = 4

Subtract 6 from both sides:

-y = -2

Multiply both sides by -1:

y = 2

So, we’ve found that y = 2.

The Solution

We've successfully solved the system of equations! Our solution is x = 6 and y = 2. This means that the point (6, 2) is the intersection of the two lines represented by these equations.

Looking back at the options provided, the correct answer is:

  • C. x = 6, y = 2

Checking Our Work

It's always a good idea to double-check our solution to make sure it works in both original equations. Let's plug in x = 6 and y = 2:

Equation 1: ½x + y = 5

½(6) + 2 = 3 + 2 = 5 (Correct!)

Equation 2: x - y = 4

6 - 2 = 4 (Correct!)

Our solution checks out! We can be confident that x = 6 and y = 2 is the correct answer.

Common Mistakes to Avoid

Solving systems of equations can be tricky, and it’s easy to make small mistakes. Here are a few common pitfalls to watch out for:

  • Sign Errors: Pay close attention to negative signs, especially when adding or subtracting equations.
  • Incorrect Substitution: Make sure you're substituting the expression for the correct variable and into the correct equation.
  • Arithmetic Errors: Double-check your calculations, especially when dealing with fractions or decimals.
  • Forgetting to Back-Substitute: Don't forget to plug the value you found back into the equation to solve for the other variable.

Tips and Tricks for Mastering Systems of Equations

  • Practice, Practice, Practice: The more you solve, the better you’ll get at recognizing patterns and choosing the best method.
  • Write Clearly: Keep your work organized and easy to follow. This helps prevent mistakes and makes it easier to check your work.
  • Check Your Solutions: Always plug your answers back into the original equations to verify that they work.
  • Visualize: If possible, graph the equations. The point of intersection is the solution to the system.
  • Choose the Best Method: Think about the structure of the equations and choose the method (substitution or elimination) that seems easiest for that particular problem.

Conclusion

Solving systems of equations might seem challenging at first, but with a clear understanding of the methods and some practice, you'll become a pro in no time! Remember, the key is to break down the problem into smaller steps, stay organized, and double-check your work. You've got this!

By understanding the elimination method, we were able to efficiently find the values of x and y that satisfy the given system of equations. Keep practicing, and you’ll become a master at solving these types of problems. Keep practicing, guys!