Solving For 't': A Step-by-Step Guide

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Hey everyone! Today, we're diving into a fundamental concept in algebra: solving for a variable. We'll specifically tackle the equation 9 = 3t + 6, where our goal is to isolate the variable 't'. Don't worry if you're feeling a little rusty – we'll break it down into easy-to-follow steps. This is a crucial skill because it forms the building block for more complex algebraic problems. Understanding how to manipulate equations and isolate variables is essential whether you're dealing with physics, economics, or even just trying to figure out a tricky puzzle. So, let's get started and make sure we all feel comfortable with this process. By the end, you'll be able to confidently solve this type of equation and apply the same methods to similar problems. This isn't just about getting an answer; it's about understanding the underlying principles of algebra! Let's get started. We're going to use basic arithmetic operations to find out the value of "t".

Step 1: Isolate the Term with 't'

Alright, guys, our first step is to get that term with 't' (which is '3t') by itself on one side of the equation. To do this, we need to get rid of the '+6' that's hanging out on the right side. The golden rule in algebra is: what you do to one side of the equation, you MUST do to the other side to keep things balanced. So, to remove '+6', we're going to subtract 6 from BOTH sides of the equation. Think of it like a seesaw – if you take weight off one side, you have to do the same on the other to keep it level. So, we start with our original equation: 9 = 3t + 6. Now, we subtract 6 from both sides. This gives us:

  • 9 - 6 = 3t + 6 - 6

When we simplify, we get:

  • 3 = 3t

See how we've successfully isolated the '3t' term on the right side? We're one step closer to solving for 't'. At this stage, it's very important to keep track of the signs (+ and -). If you make a mistake in this area, your final answer will be wrong. So keep the signs in mind, and you should be good to go. This step is about simplifying the equation by removing the constant terms from one side, to make it easier to solve for the variable. This will create a new equation that is equivalent to the original, but simpler to work with.

Step 2: Solve for 't'

Now that we have 3 = 3t, we're in the home stretch! Our goal is to get 't' completely by itself. Currently, 't' is being multiplied by 3. To undo this, we need to perform the opposite operation – division. Again, remember our rule: whatever we do to one side, we do to the other. So, we're going to divide BOTH sides of the equation by 3. This gives us:

  • 3 / 3 = 3t / 3

When we simplify, we get:

  • 1 = t

Or, if we write it the way we usually see it:

  • t = 1

And there you have it! We've solved for 't'! t equals 1. Congratulations, you've successfully solved for the variable 't' in the equation 9 = 3t + 6. That's a huge achievement! This final step involves isolating the variable by dividing both sides of the equation by the coefficient of the variable. By doing this, we get the value of 't'. Now, let's just make sure that t = 1 is correct. To do this we must put 1 in the place of t in the original equation. 9 = 3(1) + 6. Is this true? Yes! We know our answer is correct. It is always a good idea to check your work, this way you can be sure you have the correct answer. We started with the original equation and ended up with the correct answer. You can now use this approach for other similar questions. Now, let's explore a few more concepts about equation solving.

Understanding the Basics

Alright, let's back up a bit and make sure we all have a strong foundation. Solving for a variable, like 't', is all about finding the value that makes the equation true. Equations are like balanced scales; whatever you do to one side, you must do to the other to keep them balanced. The key operations we use are addition, subtraction, multiplication, and division. The goal is to isolate the variable on one side of the equation. This means getting the variable all by itself. This process relies on the concept of inverse operations. For example, to undo addition, you use subtraction; to undo multiplication, you use division. It's like a puzzle where you're trying to figure out what number fits in a particular spot. In the example of our equation, we are looking for the value of the unknown variable t. By applying the rules of algebra, we can isolate it and determine its value. This is a fundamental skill in math and can be used in many different contexts. Always be sure to keep track of your steps, and verify your answers to ensure they're correct. Let's delve a bit more into the underlying math concepts.

Important Concepts

Let's unpack some important concepts that make solving equations a breeze. First off, there's the concept of inverse operations. Inverse operations are pairs of operations that undo each other. Addition and subtraction are inverse operations. Multiplication and division are inverse operations. Understanding inverse operations is key to isolating the variable. Think of it like this: if you add something to one side, you subtract the same amount from both sides. Second, the order of operations matters. This is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). When you're solving an equation, you often need to work backward through the order of operations to isolate the variable. The third concept is the distributive property. This property allows you to multiply a term by everything inside parentheses. For instance, if you have 2(t + 3), you multiply both 't' and '3' by 2. It's really useful for simplifying equations before solving. The final concept to keep in mind is like terms. Like terms are terms that have the same variable raised to the same power. You can combine like terms by adding or subtracting their coefficients. Understanding these concepts will help you work through equations with confidence. Mastering these building blocks is essential.

Practice Problems

Now, let's get you some practice! Solving equations is like riding a bike – the more you do it, the better you get. Here are a few practice problems similar to the one we solved, along with the answers. Try these out, and then check your work to ensure you've understood the concept!

  1. 15 = 5t + 5
    • Answer: t = 2
  2. 20 = 2t - 4
    • Answer: t = 12
  3. 7 = t / 3 + 4
    • Answer: t = 9

Remember to show your work – it's the best way to catch any mistakes and reinforce your understanding. Don't worry if you don't get them right away. The main point is to keep practicing. Work through the steps we outlined earlier, and you'll become a pro in no time. If you struggle, revisit the steps, and remember the key concepts. Use these to build your confidence and become a math superstar. Practice is key, and the more you practice, the more comfortable and confident you will become. Keep up the great work!

Conclusion: You Got This!

Alright, guys, you've made it to the end! We've successfully solved for 't' in the equation 9 = 3t + 6 and explored some key concepts in the process. Remember, solving equations is a fundamental skill in algebra, and it's something you'll use throughout your math journey. Don't be afraid to practice and ask questions. The more you work with these concepts, the more comfortable and confident you'll become. Keep practicing, reviewing the steps, and most importantly, believe in yourself. You've got this, and you're well on your way to mastering algebra. Great job and keep up the amazing work.