Solving Systems Of Equations With Multiplication

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Solving Systems of Equations with Multiplication: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a cool technique to solve systems of equations: the multiplication method within the broader linear combination method. It's a handy tool that helps us find the values of x and y that satisfy two equations simultaneously. Let's break down how it works, using your example as our guide.

Understanding the Linear Combination Method

First off, what's a system of equations? It's simply a set of two or more equations, like the ones you provided:

6x - 3y = 3
-2x + 6y = 14

The goal is to find values for x and y that make both equations true at the same time. The linear combination method, also known as the elimination method, is one way to tackle this. The main idea is to manipulate the equations so that when we add or subtract them, one of the variables disappears, allowing us to solve for the other. This is where multiplication comes in super handy.

The Multiplication Trick: Why and How

Sometimes, when we look at our equations, adding or subtracting them directly won't eliminate a variable. That's when we use multiplication. We multiply one or both equations by a constant so that when we add or subtract the equations, the coefficients of either x or y become opposites (e.g., 3 and -3) or the same value, resulting in their cancellation.

Looking at your example:

6x - 3y = 3
-2x + 6y = 14

We can see that if we multiply the second equation by 3, the x terms will become 6x and -6x. This means when you add the equations together, the x variables will cancel each other out, making it easier to solve for y.

Step-by-Step Solution

Let's walk through the steps to solve this system using the multiplication method:

Step 1: Prepare the Equations

We already have our equations:

6x - 3y = 3  (Equation 1)
-2x + 6y = 14 (Equation 2)

Our goal is to eliminate either x or y. Let's choose to eliminate x. To do this, multiply Equation 2 by 3:

3 * (-2x + 6y) = 3 * 14
-6x + 18y = 42 (Modified Equation 2)

Step 2: Combine the Equations

Now we have:

6x - 3y = 3  (Equation 1)
-6x + 18y = 42 (Modified Equation 2)

Add Equation 1 and the modified Equation 2 together:

(6x - 3y) + (-6x + 18y) = 3 + 42

Step 3: Solve for the Remaining Variable

Simplify the combined equation:

6x - 6x - 3y + 18y = 45
15y = 45

Divide both sides by 15:

y = 45 / 15
y = 3

So, we found that y = 3!

Step 4: Solve for the Other Variable

Now that we know y = 3, substitute this value back into either Equation 1 or Equation 2 to solve for x. Let's use Equation 1:

6x - 3y = 3
6x - 3(3) = 3
6x - 9 = 3

Add 9 to both sides:

6x = 12

Divide both sides by 6:

x = 12 / 6
x = 2

Therefore, x = 2.

Step 5: State the Solution

The solution to the system of equations is x = 2 and y = 3. We write this as an ordered pair (2, 3).

Therefore, the answer is B. (2, 3).

Quick Recap and Tips for Success

  • Identify the Target: Decide which variable to eliminate first.
  • Multiply Strategically: Choose multipliers to create opposite or equal coefficients.
  • Combine Carefully: Add or subtract the equations to eliminate the chosen variable.
  • Solve Systematically: Solve for the remaining variable, then substitute back to find the other.
  • Double-Check: Always plug your solution back into both original equations to verify that they work.

Mastering the multiplication method is a key step towards becoming a system-solving superstar. Keep practicing, and you'll find it becomes second nature! Don't hesitate to ask if you have any questions. Happy solving!

Let's Go Through Another Example for Practice

Let's work through another example to solidify your understanding. Suppose we have the following system of equations:

2x + y = 7
x - y = 2

In this case, we can see that the y terms already have opposite coefficients (+1 and -1). This means we don't need to multiply either equation before combining them. We can directly add the two equations together.

(2x + y) + (x - y) = 7 + 2
3x = 9
x = 3

Now, substitute x = 3 back into one of the original equations. Let's use the second equation:

x - y = 2
3 - y = 2
-y = -1
y = 1

So the solution to this system is (3, 1). See how straightforward it can be when you can immediately eliminate a variable?

Advanced Scenarios and Considerations

While the method is quite simple, here are a few things to keep in mind:

  • Choosing the Best Variable: Sometimes, eliminating one variable is easier than the other. Look at the coefficients and decide which will be less effort to manipulate.
  • Fractions: If you end up with fractions, don't panic! The process is still the same. Just be careful with your arithmetic.
  • No Solution/Infinite Solutions: Some systems have no solution (the lines are parallel and never intersect) or infinite solutions (the lines are the same). You'll recognize these cases when you're solving and end up with something like 0 = 5 (no solution) or 0 = 0 (infinite solutions).

Common Mistakes to Avoid

Here are some common pitfalls:

  • Forgetting to Multiply ALL Terms: When multiplying an equation, make sure you multiply every term, including the constant term on the right side of the equation.
  • Sign Errors: Be extra careful with negative signs, especially when subtracting equations.
  • Incorrect Combining: Double-check that you're adding or subtracting the equations correctly.

Practice, Practice, Practice!

The key to mastering this method (and any math technique) is practice. Work through various examples, and don't be afraid to make mistakes—they're opportunities to learn! With each problem you solve, you'll become more confident and proficient. So, keep at it, and you'll be a pro in no time.

Final Thoughts

The multiplication method is a valuable tool in your mathematical arsenal. It provides an efficient way to solve systems of equations, which have widespread applications in various fields. By understanding and practicing this method, you equip yourself to tackle more complex problems with confidence. Keep up the great work, and happy solving!

Remember to always double-check your answer by substituting the values of x and y back into the original equations to ensure they are correct. Keep practicing, and you will become very proficient in this method. It takes a little practice but is very valuable.

Good luck, and happy problem-solving!