Solving 7x-2y=6 And 4x+4y=-3 With Elimination Method

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Hey guys! Today, we're diving into the fascinating world of solving systems of linear equations. Specifically, we're going to tackle the system:

  • 7x - 2y = 6
  • 4x + 4y = -3

using the elimination method. Buckle up, because we're about to make those variables disappear like magic!

Understanding the Elimination Method

The elimination method, also known as the addition method, is a powerful technique for solving systems of linear equations. The core idea is to manipulate the equations so that when you add them together, one of the variables cancels out, leaving you with a single equation in one variable. This makes it super easy to solve for that variable, and then you can back-substitute to find the value of the other one. Think of it as a strategic game of variable annihilation!

Before we jump into the nitty-gritty of our specific problem, let's break down the general steps involved in the elimination method:

  1. Line Up the Variables: Make sure your equations are written in the standard form (Ax + By = C). This ensures that the x and y terms are aligned, making the next steps much smoother.
  2. Multiply to Match Coefficients: This is where the magic happens! Look at the coefficients of either x or y in both equations. Our goal is to make the coefficients of one of the variables the same (but with opposite signs). To do this, you might need to multiply one or both equations by a suitable constant. The key here is to choose multipliers that will create those matching coefficients with opposite signs.
  3. Add the Equations: Once you have matching coefficients with opposite signs, add the two equations together. The chosen variable should vanish, leaving you with a single equation in one variable. It's like watching a mathematical magic trick unfold before your eyes!
  4. Solve for the Remaining Variable: Solve the resulting equation for the single variable. This is usually a straightforward algebraic step. You're one step closer to cracking the code!
  5. Substitute and Solve: Take the value you just found and substitute it back into either of the original equations. Solve this equation to find the value of the other variable. You've now found the solution to the system! Congratulations!
  6. Check Your Solution: Always, always, always check your solution by plugging the values of x and y back into both original equations. If both equations hold true, you've got it right! It's like having a secret decoder ring for equations.

Now that we have a solid grasp of the game plan, let's apply these steps to our system of equations.

Applying Elimination to 7x - 2y = 6 and 4x + 4y = -3

Let's revisit our system:

  • 7x - 2y = 6
  • 4x + 4y = -3

Step 1: Line Up the Variables

Good news! Our equations are already in the standard form (Ax + By = C), so the x and y terms are nicely aligned. We're off to a great start!

Step 2: Multiply to Match Coefficients

Now, let's strategize. Looking at the y coefficients, we have -2 and 4. The least common multiple of 2 and 4 is 4. To make the y coefficients match with opposite signs, we can multiply the first equation by 2. This will give us -4y in the first equation and +4y in the second equation – perfect for elimination!

Multiplying the first equation (7x - 2y = 6) by 2, we get:

2 * (7x - 2y) = 2 * 6
14x - 4y = 12

Our system now looks like this:

  • 14x - 4y = 12
  • 4x + 4y = -3

See how the y coefficients are -4 and +4? We're ready for the next step!

Step 3: Add the Equations

Now comes the satisfying part – adding the equations together. Let's add the left-hand sides and the right-hand sides separately:

(14x - 4y) + (4x + 4y) = 12 + (-3)

Simplifying, we get:

18x = 9

The y terms have vanished! It's like they never existed. We're left with a simple equation in x.

Step 4: Solve for the Remaining Variable

To solve for x, divide both sides of the equation by 18:

18x / 18 = 9 / 18
x = 1/2

We've found the value of x! x = 1/2. We're halfway there!

Step 5: Substitute and Solve

Now, let's substitute x = 1/2 back into one of the original equations to find y. We can choose either equation, but let's go with the first one (7x - 2y = 6) because it looks a little simpler.

Substituting x = 1/2, we get:

7 * (1/2) - 2y = 6
7/2 - 2y = 6

To solve for y, first subtract 7/2 from both sides:

-2y = 6 - 7/2
-2y = 12/2 - 7/2
-2y = 5/2

Now, divide both sides by -2:

y = (5/2) / (-2)
y = -5/4

We've found the value of y! y = -5/4.

Step 6: Check Your Solution

Time for the final boss – checking our solution! Let's plug x = 1/2 and y = -5/4 back into both original equations to make sure they hold true.

  • Equation 1: 7x - 2y = 6

    7 * (1/2) - 2 * (-5/4) = 6
7/2 + 10/4 = 6
7/2 + 5/2 = 6
12/2 = 6
6 = 6  (True!)

  • Equation 2: 4x + 4y = -3

    4 * (1/2) + 4 * (-5/4) = -3
2 - 5 = -3
-3 = -3  (True!)

Both equations hold true! We've successfully solved the system of equations.

Solution

The solution to the system of equations 7x - 2y = 6 and 4x + 4y = -3 is:

  • x = 1/2
  • y = -5/4

We can also write this as an ordered pair: (1/2, -5/4).

Key Takeaways

  • The elimination method is a powerful tool for solving systems of linear equations.
  • The key is to manipulate the equations so that one variable cancels out when you add them together.
  • Always check your solution to ensure accuracy.
  • Solving systems of equations is a fundamental skill in algebra and has applications in various fields, from engineering to economics.

Practice Makes Perfect

The best way to master the elimination method is to practice, practice, practice! Try solving different systems of equations with varying coefficients and complexities. Don't be afraid to make mistakes – they're learning opportunities in disguise. Keep at it, and you'll become a system-solving pro in no time!

Conclusion

So, there you have it, folks! We've successfully tackled the system of equations 7x - 2y = 6 and 4x + 4y = -3 using the elimination method. Remember the steps, practice diligently, and you'll be solving linear equations like a champ. Keep exploring the fascinating world of math, and I'll catch you in the next one! Remember that math is like a puzzle, and each equation is a piece. The elimination method helps you fit those pieces together to reveal the solution. This method isn't just about getting the right answer; it's about understanding the process and developing your problem-solving skills. Mastering these skills will not only help you in your math classes but also in various aspects of your life where logical thinking and analytical abilities are crucial.

One common mistake students make is forgetting to multiply every term in the equation when multiplying by a constant. Always remember to distribute the multiplication across all terms to maintain the equation's balance. Another tricky part can be choosing the right multiplier. Look for the least common multiple of the coefficients to make your calculations easier. Sometimes, you might need to multiply both equations to get the coefficients to match, but with practice, you'll develop an intuition for the most efficient approach. Keep challenging yourself with more complex systems, and you'll soon find that you can conquer any linear equation that comes your way! Remember, the journey of learning math is like climbing a mountain. There will be challenges along the way, but the view from the top – the understanding and the satisfaction of solving a tough problem – is worth the effort. So, keep climbing, keep exploring, and keep solving!