Multiplying Binomials: (4x - Y)(5x - 7y) Simplified

Alright guys, let's dive into some algebra and tackle the problem of multiplying and simplifying the expression (4x - y)(5x - 7y). This might seem a bit daunting at first, but don't worry! We'll break it down step by step, making it super easy to understand. We're going to use a method that many find helpful, often referred to as the FOIL method. This acronym stands for First, Outer, Inner, Last, and it's a systematic way to ensure we multiply each term in the first binomial by each term in the second binomial. Trust me, once you get the hang of this, you'll be multiplying binomials like a pro!

Understanding the FOIL Method

Before we jump into the actual multiplication, let's quickly recap what the FOIL method is all about. It's essentially a mnemonic device to help us remember the order in which to multiply the terms of two binomials. Think of it as a checklist to ensure we don't miss anything. Here’s a breakdown:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outer terms in the expression.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms in each binomial.

By following this order, we can ensure we've covered all the bases and multiplied every term correctly. This is especially important when dealing with more complex expressions, as it helps keep our work organized and reduces the chance of making errors. So, with the FOIL method fresh in our minds, let's get started on simplifying our expression (4x - y)(5x - 7y). Remember, math is like building blocks – each step builds upon the previous one, so let's lay a solid foundation!

Applying FOIL: Step-by-Step

Okay, let’s get our hands dirty and apply the FOIL method to our expression, (4x - y)(5x - 7y). We'll go through each step meticulously, so you can see exactly how it works.

1. First: We multiply the first terms in each binomial. These are 4x and 5x. So, (4x) * (5x) = 20x². Remember, when we multiply variables with exponents, we add the exponents. In this case, x has an exponent of 1 (which we usually don't write), so x¹ * x¹ = x¹⁺¹ = x².
2. Outer: Next up are the outer terms, which are 4x and -7y. Multiplying these gives us (4x) * (-7y) = -28xy. Pay close attention to the signs here! A positive times a negative results in a negative.
3. Inner: Now we move on to the inner terms, -y and 5x. When we multiply these, we get (-y) * (5x) = -5xy. Again, watch those signs!
4. Last: Finally, we multiply the last terms in each binomial, -y and -7y. This gives us (-y) * (-7y) = 7y². A negative times a negative equals a positive, so we end up with a positive term.

So far, we've expanded our expression using the FOIL method. We've multiplied each term correctly and kept track of the signs. Now, we have a longer expression: 20x² - 28xy - 5xy + 7y². But we're not done yet! The next step is to simplify this by combining like terms. This is where our expression starts to look cleaner and more manageable. Stick with me, guys – we're almost there!

Combining Like Terms

Now that we've applied the FOIL method, we have the expanded expression: 20x² - 28xy - 5xy + 7y². The next crucial step is to simplify this expression by combining like terms. This is like tidying up your room – we're grouping similar items together to make things neater and easier to understand. In algebraic terms, like terms are those that have the same variables raised to the same powers. For example, -28xy and -5xy are like terms because they both have the variables x and y, each raised to the power of 1.

Looking at our expression, we can see that we have two terms that contain both x and y: -28xy and -5xy. These are the like terms we need to combine. To do this, we simply add their coefficients (the numbers in front of the variables). So, -28xy + (-5xy) = -33xy. Remember, when adding negative numbers, we're essentially moving further into the negative direction on the number line.

The other terms in our expression, 20x² and 7y², don't have any like terms. This means they stay as they are. They're unique in our expression, and there's nothing else to combine them with. Think of them as the lone wolves of our equation!

So, after combining like terms, our expression transforms from 20x² - 28xy - 5xy + 7y² to a much simpler form. We've taken a slightly messy expression and streamlined it into something much more elegant. This is a key skill in algebra, as it allows us to work with expressions more efficiently. So, what's our simplified expression? Let's put it all together and see!

The Simplified Expression

After meticulously applying the FOIL method and then combining like terms, we've finally arrived at the simplified form of our expression. Remember, we started with (4x - y)(5x - 7y), and after a bit of algebraic maneuvering, we've transformed it into something much cleaner and easier to work with.

So, let's reveal the grand finale! Combining the results of our previous steps, the simplified expression is:

20x² - 33xy + 7y²

And there you have it! We've successfully multiplied the binomials and simplified the result. Notice how much cleaner and more concise this expression is compared to the expanded form we had after applying the FOIL method. This simplified form is not only more aesthetically pleasing, but it's also much more practical for further algebraic manipulations, like solving equations or graphing. You guys rock!

Why is Simplifying Important?

You might be wondering,