Multiply (x-y+4) And (3x-1) With Verification

by Sebastian Müller 46 views

Hey guys! Let's dive into a fun math problem today. We're going to tackle multiplying two expressions: (x - y + 4) and (3x - 1). But that's not all! We'll also verify our result by plugging in some specific values for x and y. So, buckle up and let's get started!

Expanding the Expressions: Unleashing the Power of Distribution

First things first, we need to multiply these expressions. To do this, we'll use the distributive property, which is basically like sharing the love (or the multiplication, in this case) to each term inside the parentheses. It might sound intimidating, but trust me, it's easier than it looks. We're essentially going to multiply each term in the first expression (x - y + 4) by each term in the second expression (3x - 1).

Let's break it down step by step:

  1. Multiply x by the entire second expression (3x - 1):

    • x * (3x - 1) = x * 3x + x * (-1) = 3x² - x

  2. Multiply -y by the entire second expression (3x - 1):

    • -y * (3x - 1) = -y * 3x + (-y) * (-1) = -3xy + y

  3. Multiply 4 by the entire second expression (3x - 1):

    • 4 * (3x - 1) = 4 * 3x + 4 * (-1) = 12x - 4

Now, we've distributed everything! But we're not done yet. We need to combine all the terms we've obtained. So, let's put them all together:

3x² - x - 3xy + y + 12x - 4

Simplifying the Result: Taming the Algebraic Jungle

Okay, we've got a bunch of terms here. The next step is to simplify our expression by combining any like terms. Like terms are those that have the same variable raised to the same power. In our expression, we have a couple of like terms involving x:

  • -x and +12x

Let's combine these guys:

-x + 12x = 11x

Now, let's rewrite our entire expression with the simplified terms:

3x² - 3xy + 11x + y - 4

And there you have it! This is our simplified expression after multiplying (x - y + 4) and (3x - 1). We've successfully navigated the algebraic jungle and emerged victorious! This is our expanded and simplified form. It's crucial to take your time and double-check each step to avoid any silly mistakes. Math is like a puzzle, and each piece needs to fit perfectly.

Verifying the Result: The Moment of Truth

Now comes the exciting part – verifying our result! This is where we get to plug in some specific values for x and y and see if our simplified expression holds true. The problem tells us to verify for x = 2 and y = -1. This is a great way to ensure we haven't made any errors along the way. If our verification works out, we can be confident in our solution.

Plugging in the Values: A Numerical Adventure

First, we'll plug these values into our original expressions:

(x - y + 4) becomes (2 - (-1) + 4) (3x - 1) becomes (3(2) - 1)

Let's simplify these:

(2 - (-1) + 4) = (2 + 1 + 4) = 7 (3(2) - 1) = (6 - 1) = 5

So, the product of the original expressions with these values is:

7 * 5 = 35

Now, let's plug the same values into our simplified expression:

3x² - 3xy + 11x + y - 4 becomes 3(2)² - 3(2)(-1) + 11(2) + (-1) - 4

Evaluating the Simplified Expression: The Final Showdown

Let's simplify this step by step, following the order of operations (PEMDAS/BODMAS):

  1. Exponents: 3(2)² = 3 * 4 = 12
  2. Multiplication:
    • -3(2)(-1) = -6(-1) = 6
    • 11(2) = 22

Now, let's put it all together:

12 + 6 + 22 - 1 - 4

  1. Addition and Subtraction (from left to right):
    • 12 + 6 = 18
    • 18 + 22 = 40
    • 40 - 1 = 39
    • 39 - 4 = 35

Ta-da! We got 35! This is the same result we obtained when we plugged the values into the original expressions. This confirms that our simplified expression is correct. We've successfully verified our work! It's always a good feeling when the numbers align and everything checks out.

Conclusion: Mastering the Art of Multiplication and Verification

So, there you have it! We've successfully multiplied the expressions (x - y + 4) and (3x - 1), simplified the result, and verified our answer by plugging in specific values. We've shown that the expanded and simplified form of (x - y + 4)(3x - 1) is 3x² - 3xy + 11x + y - 4. Remember, the key to mastering these kinds of problems is to take it step by step, be organized, and double-check your work. And most importantly, don't be afraid to make mistakes – they're just opportunities to learn! Keep practicing, and you'll become a math whiz in no time!

Remember, math isn't just about getting the right answer; it's about understanding the process and building your problem-solving skills. So, keep exploring, keep learning, and keep having fun with math! You've got this!