Simplify Algebraic Expressions: Step-by-Step Solutions
Hey guys! Let's break down some algebra problems together. We're going to simplify expressions, which basically means cleaning them up to look their best. Think of it like tidying up your room – you want everything neat and in its place. In algebra, that means combining like terms. Like terms are terms that have the same variable raised to the same power. For example, 2x and 7x are like terms because they both have x to the power of 1. Similarly, 3y and 5y are like terms. But 2x and 3y are not like terms because they have different variables. And 2x and 3x² are not like terms because they have the same variable but different powers.
1. Combining Like Terms: 2x + 3y + 7x + 5y
Let's start with the first expression: 2x + 3y + 7x + 5y. The key here is to identify and combine the like terms.
- First, group the 'x' terms together:
2x + 7x - Then, group the 'y' terms together:
3y + 5y
Now, add the coefficients (the numbers in front of the variables) of the like terms:
2x + 7x = (2 + 7)x = 9x3y + 5y = (3 + 5)y = 8y
So, the simplified expression is 9x + 8y. That's it! We've combined all the like terms and made the expression as simple as possible. Remember, you can only add or subtract like terms. You can't combine x and y terms because they're different variables. Think of it like trying to add apples and oranges – you can't combine them into a single type of fruit, you just have apples and oranges.
This principle extends to more complex expressions as well. For instance, if you had 2x² + 3x + 5x² - x, you would combine the x² terms (2x² + 5x² = 7x²) and the x terms (3x - x = 2x) to get the simplified expression 7x² + 2x. Always look for the terms with the exact same variable and power before combining. This is the fundamental rule of simplifying algebraic expressions, and mastering it will make algebra much easier. So, take your time, practice identifying like terms, and you'll be simplifying expressions like a pro in no time!
2. Simplifying with Negative Coefficients: -4a + 8b - 2a - 5b
Next up, we have the expression -4a + 8b - 2a - 5b. Don't let the negative signs scare you! The process is exactly the same: identify and combine like terms. A negative coefficient simply means you're subtracting that term.
- Group the 'a' terms:
-4a - 2a - Group the 'b' terms:
8b - 5b
Now, combine the coefficients:
-4a - 2a = (-4 - 2)a = -6a8b - 5b = (8 - 5)b = 3b
Therefore, the simplified expression is -6a + 3b. See? The negative signs just come along for the ride. The key is to pay attention to the sign before the term. That sign belongs to that term.
Consider a slightly more complex example, such as -7x + 3y - 2x + 5y - x. Here, you have multiple 'x' and 'y' terms to combine. First, combine all the 'x' terms: -7x - 2x - x = (-7 - 2 - 1)x = -10x. Then, combine all the 'y' terms: 3y + 5y = (3 + 5)y = 8y. The simplified expression is -10x + 8y. The process is always the same, regardless of the number of terms or the presence of negative signs.
Remember to treat each term individually, paying close attention to its sign and variable. If you keep practicing, these types of problems will become second nature. With practice, you will quickly and accurately identify like terms and combine their coefficients, even when they include negative numbers. This is a crucial skill for success in algebra and beyond!
3. Simplifying Squared Terms: 5a² + a²
Now, let's tackle 5a² + a². This might look a little different because of the exponent (the little 2), but the principle is the same. Remember, a² means 'a squared' or 'a to the power of 2'. It's simply a * a.
The important thing is that both terms have a². They are like terms! Think of a² as a single unit. It's like saying you have 5 apples plus 1 apple.
So, we have:
5a² + a² = (5 + 1)a² = 6a²
The simplified expression is 6a². Easy peasy!
Let's try a similar example: 12b² - 4b². Both terms have b², so they are like terms. We simply subtract the coefficients: 12b² - 4b² = (12 - 4)b² = 8b². The simplified expression is 8b². The same rule applies regardless of the coefficient values. For instance, 25x² + 15x² = (25 + 15)x² = 40x². As long as the variable and its exponent are the same, you can combine the terms by adding or subtracting their coefficients.
Keep in mind that you cannot combine a² with a. They are not like terms because the exponents are different. a² is a * a, while a is just a. They represent different quantities. Therefore, an expression like 5a² + a cannot be simplified further. It's already in its simplest form. Understanding the concept of like terms is essential for correctly simplifying algebraic expressions involving exponents. So, always double-check that the variables and their exponents are identical before combining the terms!
4. Combining Polynomials: 3x² - 6x + 1 - 2x² + 4x
This expression, 3x² - 6x + 1 - 2x² + 4x, looks a bit longer, but don't worry! We'll just take it one step at a time. It's a polynomial, which just means it has multiple terms with different powers of x.
Again, we need to identify and combine like terms. This time, we have three different types of terms: x² terms, x terms, and constant terms (numbers without any variables).
- Group the
x²terms:3x² - 2x² - Group the
xterms:-6x + 4x - The constant term is just
1(it doesn't have any like terms to combine with)
Now, combine the like terms:
3x² - 2x² = (3 - 2)x² = 1x² = x²-6x + 4x = (-6 + 4)x = -2x
So, the simplified expression is x² - 2x + 1. Remember to write the terms in descending order of their exponents (highest power first).
Let's consider a slightly more complex example: 4y³ - 2y² + 5y - 7 + y³ + 3y² - 2y + 3. Here, we have y³, y², y, and constant terms. Grouping like terms gives us (4y³ + y³) + (-2y² + 3y²) + (5y - 2y) + (-7 + 3). Combining these gives us 5y³ + y² + 3y - 4. Remember to pay attention to the signs and combine the coefficients carefully. With practice, simplifying polynomials will become a straightforward process.
Remember that the order of operations does not affect how you combine like terms. You can rearrange the terms as needed to group them together, as long as you keep the signs consistent. The key is to accurately identify the like terms and combine their coefficients. This skill is fundamental for solving more complex algebraic equations and problems, so mastering it is a worthwhile investment of your time and effort!
5. Simplifying with Parentheses: (7a + b) + (-9a + 8b)
Finally, let's look at (7a + b) + (-9a + 8b). The parentheses might look intimidating, but in this case, they're actually not doing much. Because we're adding the two expressions inside the parentheses, we can simply remove them.
So, we have:
7a + b - 9a + 8b
Now, combine like terms as usual:
- Group the 'a' terms:
7a - 9a - Group the 'b' terms:
b + 8b
Combine the coefficients:
7a - 9a = (7 - 9)a = -2ab + 8b = (1 + 8)b = 9b
The simplified expression is -2a + 9b. And we're done!
However, it is important to note that the presence of a minus sign before a set of parentheses will change the signs of all the terms inside the parentheses. For instance, in the expression (3x + 2y) - (x - y), the minus sign before the second set of parentheses requires us to distribute the negative sign to each term inside. This means we change the signs: 3x + 2y - x + y. Combining like terms then gives us 2x + 3y. Always be mindful of the sign before the parentheses and remember to distribute it correctly to ensure you simplify the expression accurately.
Another example of simplifying expressions with parentheses is 2(a + 3b) + 4(2a - b). Here, you need to distribute the numbers outside the parentheses to each term inside: 2a + 6b + 8a - 4b. Then, combine like terms: (2a + 8a) + (6b - 4b) = 10a + 2b. So, the simplified expression is 10a + 2b. Mastering the ability to simplify expressions with parentheses is a crucial skill for success in algebra, allowing you to solve more complex equations and problems with confidence.
Simplifying algebraic expressions is a fundamental skill in algebra. By combining like terms, you can make expressions easier to understand and work with. Remember to pay attention to the signs of the terms and to distribute any coefficients correctly. With practice, you'll be able to simplify even the most complex expressions with ease!