Solving 2x + Y = 5 And 4x + 2y = 10 With The Mixed Method

Hey guys! Ever stumbled upon a system of equations that looks a bit intimidating? Don't worry, we've all been there! Today, we're going to break down a classic example: 2x + y = 5 and 4x + 2y = 10. We'll tackle this using the mixed method, which is a super handy way to solve these kinds of problems. So, grab your pencils, and let's dive in!

Understanding the Mixed Method

So, what exactly is the mixed method? Well, it's a combination of two powerful techniques: substitution and elimination. Think of it as having the best of both worlds! Sometimes, substitution is easier for one equation, while elimination works better for another. The mixed method lets us pick the most efficient approach for each part of the system, making the whole process smoother and less prone to errors. It’s like choosing the right tool for the right job – a key strategy in problem-solving!

Why Use the Mixed Method?

You might be wondering, “Why not just stick to one method?” That’s a fair question! The mixed method shines when dealing with systems where one equation is easily solved for a single variable (perfect for substitution), while another equation has coefficients that make elimination a breeze. For example, if one equation has a lone y term, solving for y and substituting it into the other equation can be quicker than trying to eliminate variables directly. On the other hand, if you notice that multiplying one equation by a constant will make the coefficients of either x or y match in both equations, elimination becomes a very attractive option. Using the mixed method allows us to optimize our approach and avoid unnecessary complications, saving us time and effort in the long run.

The Power of Flexibility

The true beauty of the mixed method lies in its flexibility. It empowers us to analyze the system of equations and choose the most strategic path to the solution. This not only makes the process more efficient but also deepens our understanding of the underlying concepts. By recognizing the strengths of both substitution and elimination, we become more versatile problem-solvers, capable of tackling a wider range of mathematical challenges with confidence. Remember, math isn't just about memorizing formulas; it's about developing a toolkit of techniques and knowing when to use each one. The mixed method is a valuable addition to that toolkit!

Applying the Mixed Method to Our Equations

Okay, enough theory! Let's get our hands dirty and apply the mixed method to our specific problem: 2x + y = 5 and 4x + 2y = 10. Looking at these equations, which method seems like a good starting point? Well, the first equation, 2x + y = 5, has a lone y term, which makes it perfect for solving for y. Let's do that!

Step 1: Solve for y in the First Equation

To isolate y, we simply subtract 2x from both sides of the equation:

2x + y - 2x = 5 - 2x

This simplifies to:

y = 5 - 2x

Awesome! We've now expressed y in terms of x. This is the first key piece of our puzzle.

Step 2: Substitute into the Second Equation

Now that we have y = 5 - 2x, we can substitute this expression into the second equation, 4x + 2y = 10. This is where the substitution magic happens!

Replace y with (5 - 2x) in the second equation:

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

Now, we have an equation with only one variable, x. This is a huge step forward!

Step 3: Simplify and Solve for x

Let's simplify the equation by distributing the 2:

4x + 10 - 4x = 10

Notice anything interesting? The 4x and -4x terms cancel each other out:

10 = 10

Whoa! We're left with a statement that's always true. What does this mean?

Interpreting the Result: Infinite Solutions

When we end up with a true statement like 10 = 10, it tells us that our system of equations has infinitely many solutions. This means that the two equations actually represent the same line! Think of it as two paths that perfectly overlap – there are countless points where they intersect.

Understanding Dependent Systems

This type of system is called a dependent system. The equations are dependent because one equation is simply a multiple of the other. In our case, if you multiply the first equation (2x + y = 5) by 2, you get the second equation (4x + 2y = 10). This is a clear sign that we're dealing with a dependent system and, therefore, infinite solutions.

Expressing the Solutions

Since we have infinite solutions, we can't list them all out individually. Instead, we express the solutions in terms of one of the variables. We already have y = 5 - 2x. This equation represents all the solutions to the system. For any value we choose for x, we can plug it into this equation to find the corresponding value of y. For example:

• If x = 0, then y = 5 - 2(0) = 5
• If x = 1, then y = 5 - 2(1) = 3
• If x = 2, then y = 5 - 2(2) = 1

And so on! There are endless possibilities.

Checking Our Work (Always a Good Idea!)

Even though we've arrived at our answer, it's always a smart move to check our work. Let's pick a couple of solutions we found and plug them back into the original equations to make sure they hold true.

Checking the Solution (0, 5)

• Equation 1: 2x + y = 5
  • 2(0) + 5 = 5
  • 0 + 5 = 5
  • 5 = 5 (True!)
• Equation 2: 4x + 2y = 10
  • 4(0) + 2(5) = 10
  • 0 + 10 = 10
  • 10 = 10 (True!)

Checking the Solution (1, 3)

• Equation 1: 2x + y = 5
  • 2(1) + 3 = 5
  • 2 + 3 = 5
  • 5 = 5 (True!)
• Equation 2: 4x + 2y = 10
  • 4(1) + 2(3) = 10
  • 4 + 6 = 10
  • 10 = 10 (True!)

Both solutions work! This gives us confidence that our answer is correct.

Conclusion: Mastering the Mixed Method

So, there you have it! We successfully solved the system of equations 2x + y = 5 and 4x + 2y = 10 using the mixed method. We discovered that the system has infinitely many solutions because the equations are dependent. Remember, the mixed method is a powerful tool that combines the strengths of substitution and elimination, allowing us to tackle a wide variety of systems with confidence. Keep practicing, and you'll become a master of equation-solving in no time! Keep up the great work, guys!