Solving Equations & Simplifying Expressions: A Math Guide

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Hey guys! Let's dive into some math problems. This guide will help you understand how to simplify expressions and solve equations. We'll break down the problems step-by-step so it's super easy to follow along. So grab your pens and paper, and let's get started!

3a. Simplifying Expressions: (2x + 3y) + (5x - y)

Alright, let's start with simplifying the expression (2x + 3y) + (5x - y). Simplifying expressions is all about combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, our like terms are the 'x' terms and the 'y' terms. Here's how we do it:

  1. Identify Like Terms:

    • We have 2x and 5x (both have 'x').
    • We have 3y and -y (both have 'y').

  2. Combine Like Terms:

    • Combine the 'x' terms: 2x + 5x = 7x
    • Combine the 'y' terms: 3y - y = 2y (Remember, -y is the same as -1y)

  3. Write the Simplified Expression:

    • Put the combined terms together: 7x + 2y

So, the simplified form of (2x + 3y) + (5x - y) is 7x + 2y. See? Not too bad, right? The key here is to carefully identify the like terms and then add or subtract their coefficients (the numbers in front of the variables). Pay close attention to the signs (+ or -) in front of each term. That's crucial! If you're feeling a bit rusty, you might want to review basic addition and subtraction rules with positive and negative numbers. This is a fundamental concept in algebra, so making sure you're comfortable with it will make your life much easier down the road. Remember, practice makes perfect! Try creating your own similar expressions and simplifying them to build your confidence. You could even challenge yourself by including more terms or different variables. This way, you'll become a pro in no time! Keep in mind, when you are combining like terms, you're essentially just counting how many of each variable you have. Think of 'x' as apples and 'y' as bananas. If you have 2 apples and someone gives you 5 more apples, you now have 7 apples. The same logic applies to variables. Finally, always double-check your work to avoid making simple mistakes. It's easy to get mixed up, especially when dealing with negative signs. Take your time, stay focused, and you'll nail it!

3b. Simplifying Expressions: (4a - 2b) - (a + b)

Now, let's tackle (4a - 2b) - (a + b). This problem introduces subtraction, so we need to be extra careful with the signs. The core principle remains the same: combine like terms.

  1. Distribute the Negative Sign:

    • The minus sign in front of the parentheses means we need to subtract everything inside the parentheses (a + b). This is the same as multiplying each term inside the parentheses by -1.
    • So, -(a + b) becomes -a - b.

  2. Rewrite the Expression:

    • Our expression now becomes 4a - 2b - a - b.

  3. Identify Like Terms:

    • We have 4a and -a (both have 'a').
    • We have -2b and -b (both have 'b').

  4. Combine Like Terms:

    • Combine the 'a' terms: 4a - a = 3a
    • Combine the 'b' terms: -2b - b = -3b (Remember, -b is the same as -1b)

  5. Write the Simplified Expression:

    • Put the combined terms together: 3a - 3b

Therefore, the simplified form of (4a - 2b) - (a + b) is 3a - 3b. Great job! The trickiest part here is remembering to distribute the negative sign correctly. Always make sure you're subtracting each term inside the parentheses. If you're unsure, write it out step-by-step: -1 * a = -a and -1 * b = -b. Always take the time to double-check your sign. You can also think of subtraction as adding the opposite. So, 4a - 2b - (a + b) is the same as 4a - 2b + (-a - b). This perspective can sometimes make it easier to visualize the problem, especially when negative signs are involved. Also, remember to stay organized! Write neatly, keep track of your work, and use a separate sheet of paper for calculations if needed. This will minimize the chances of making careless mistakes. Practice is your best friend. Work through various examples, varying the coefficients and signs, to solidify your grasp on the concept. Soon, you'll become proficient at handling these types of problems with ease.

3c. Expanding Expressions: 2(3m + 4n)

Alright, let's switch gears and move on to expanding expressions! We'll begin with 2(3m + 4n). Expanding means removing the parentheses by multiplying the term outside the parentheses by each term inside the parentheses. This is all about applying the distributive property.

  1. Multiply:

    • Multiply the term outside the parentheses (2) by each term inside the parentheses.
    • 2 * 3m = 6m
    • 2 * 4n = 8n

  2. Write the Expanded Expression:

    • Put the results together: 6m + 8n

So, the expanded form of 2(3m + 4n) is 6m + 8n. See how easy that was? The key here is to remember to multiply every term inside the parentheses. Don't miss any! Make sure you pay attention to the signs as well. If the term outside the parentheses is negative, then the signs of the terms inside the parentheses will change accordingly. For example, if we had -2(3m + 4n), then the expanded form would be -6m - 8n. Always double-check your multiplication! It's easy to make a mistake, especially if you're working with larger numbers or fractions. Consider writing out the multiplication step-by-step to avoid errors. Try working backward: can you factor your result? Does it match the original expression? This will help you verify your work. Also, try creating your own expressions and expanding them. Practice with different coefficients, signs, and variables. The more you practice, the more confident you'll become. Remember to take your time and stay focused. Don't rush through the steps, and always double-check your calculations. Soon, you'll be expanding expressions like a pro! Break the problem down into smaller, manageable steps. This will make the entire process more straightforward and less prone to errors. Remember that the distributive property is a fundamental tool in algebra, so understanding it well is essential for success.

3d. Expanding Expressions: (x + 2)(x - 3)

Let's try a slightly more complex expansion: (x + 2)(x - 3). This involves multiplying two binomials (expressions with two terms). We'll use the FOIL method (First, Outer, Inner, Last) to help us.

  1. FOIL Method:

    • First: Multiply the first terms of each binomial: x * x = x^2
    • Outer: Multiply the outer terms: x * -3 = -3x
    • Inner: Multiply the inner terms: 2 * x = 2x
    • Last: Multiply the last terms: 2 * -3 = -6

  2. Combine Like Terms:

    • We have -3x and 2x as like terms.
    • Combine them: -3x + 2x = -x

  3. Write the Expanded Expression:

    • Put all the terms together: x^2 - x - 6

Therefore, the expanded form of (x + 2)(x - 3) is x^2 - x - 6. Using the FOIL method is a very handy trick for multiplying binomials. This helps ensure that you multiply every term correctly. Make sure you understand why the FOIL method works. Essentially, you're distributing each term in the first binomial to each term in the second binomial. Another strategy you can employ is to draw arrows to indicate which terms you're multiplying. This visual cue can help you stay organized and avoid missing any multiplications. Always pay close attention to the signs. A negative sign can easily lead to an error if you're not careful. Also, remember to combine like terms after you've multiplied all the terms. Often, there will be terms that can be simplified. Practice is absolutely critical here. Work through various examples with different variables, coefficients, and signs. You can also check your work by substituting a value for 'x' into both the original expression and the expanded form. If both expressions result in the same value, it's a good sign that you've done the expansion correctly. Try to remember that the order in which you perform the multiplications doesn't affect the result, provided you multiply every possible pair of terms. This understanding will become even more useful as you advance to more complex algebra problems.

3e. Solving Equations: 3x + 5 = 14

Alright, let's solve an equation! We will solve the equation: 3x + 5 = 14. Solving an equation is about isolating the variable (in this case, 'x') on one side of the equation. Here's how to do it:

  1. Isolate the term with x:

    • To get the term with 'x' (3x) by itself, we need to get rid of the +5.
    • Subtract 5 from both sides of the equation (to keep the equation balanced): 3x + 5 - 5 = 14 - 5
    • This simplifies to 3x = 9

  2. Solve for x:

    • To get 'x' by itself, we need to divide both sides by 3: 3x / 3 = 9 / 3
    • This simplifies to x = 3

So, the solution to the equation 3x + 5 = 14 is x = 3. You did it! The key here is remembering that whatever you do to one side of the equation, you must do to the other side to keep it balanced. This fundamental principle ensures that your solution remains correct. When solving equations, always work to isolate the variable. Start by addressing terms that are added or subtracted, and then move on to terms that are multiplied or divided. Take your time and be careful with your calculations. Double-check your arithmetic, especially when dealing with negative numbers. You can always check your answer by substituting it back into the original equation. For example, in our case, we can substitute x = 3 into 3x + 5 = 14 to see if it holds true: 3(3) + 5 = 9 + 5 = 14. Since this is correct, we know that our solution is accurate. Remember that solving equations is an essential skill in mathematics, so consistent practice is important. Try creating your own equations and solving them. Start with simpler equations and gradually increase the difficulty. Remember, practice makes perfect. The more you solve equations, the more confident and comfortable you will become. You can also explore different types of equations, such as equations with fractions or parentheses, to broaden your understanding and skills. Remember to always double-check your work to avoid making simple mistakes. Keep practicing, and you'll become a pro at solving equations!