Solving For T: A Step-by-Step Guide To The Equation

by Andrew McMorgan 52 views

Hey math enthusiasts! Ever get stumped by an equation and feel like you're staring at a jumbled mess of numbers and variables? Don't worry, we've all been there! Today, we're going to break down a common type of algebraic equation and show you how to solve for the variable t. We'll take it step-by-step, so even if you're just starting out with algebra, you'll be able to follow along. So, let's dive into the world of equations and learn how to conquer them!

Understanding the Equation: (1/8)t - 5 + (1/4)t = t + 5

Before we jump into the solution, let's take a closer look at the equation itself: (1/8)t - 5 + (1/4)t = t + 5. At first glance, it might seem a little intimidating, but don't let those fractions scare you! The key here is to understand what each part of the equation represents and what our ultimate goal is: to isolate t on one side of the equation. So, what do we mean by isolating t? Well, it simply means getting t by itself, so we have an equation that looks like t = (some number). That number will be our solution!

Breaking it Down Piece by Piece

  • (1/8)t: This term represents one-eighth of the variable t. Remember that in algebra, when a number is written next to a variable, it means we're multiplying them. So, (1/8)t is the same as (1/8) * t.
  • - 5: This is a constant term, meaning it's a number that doesn't change. It's being subtracted from the term (1/8)t.
  • + (1/4)t: This term represents one-quarter of the variable t, and it's being added to the expression.
  • =: This is the equals sign, and it's the heart of our equation. It tells us that the expression on the left side of the sign is equal in value to the expression on the right side.
  • t: This is the variable we're trying to solve for. It appears on both sides of the equation, which is common in algebra problems.
  • + 5: This is another constant term, and it's being added to the variable t on the right side of the equation.

Our mission, should we choose to accept it (and we do!), is to manipulate this equation using algebraic rules until we have t all alone on one side. Think of it like a balancing act: whatever we do to one side of the equation, we must also do to the other side to keep things balanced. Let's get started!

Step 1: Combining Like Terms on the Left Side

Okay, the first thing we want to do is simplify the equation by combining like terms. Like terms are terms that have the same variable raised to the same power. In our equation, (1/8)t - 5 + (1/4)t = t + 5, we have two terms with the variable t: (1/8)t and (1/4)t. These are like terms, and we can combine them.

To combine them, we need to add their coefficients. The coefficient is the number that's being multiplied by the variable. So, we need to add 1/8 and 1/4. But how do we add fractions? We need a common denominator!

  • The least common denominator (LCD) of 8 and 4 is 8. So, we need to convert 1/4 to an equivalent fraction with a denominator of 8. To do this, we multiply both the numerator and the denominator of 1/4 by 2: (1 * 2) / (4 * 2) = 2/8.
  • Now we can add the fractions: 1/8 + 2/8 = 3/8

So, (1/8)t + (1/4)t = (3/8)t. Now we can rewrite our equation with the combined like terms:

(3/8)t - 5 = t + 5

See? We've already made progress! The equation looks a little cleaner now. We've taken those two t terms and combined them into one. This is a crucial step in solving for t, as it simplifies the equation and makes it easier to work with. Remember, the goal is to isolate t, and by combining like terms, we're one step closer to achieving that goal.

Step 2: Moving the Variable Terms to One Side

Alright, we've combined the like terms on the left side, and now it's time to get all the t terms together on one side of the equation. Currently, we have (3/8)t on the left and t on the right. To get all the t terms on the left, we need to get rid of the t on the right side. How do we do that? By using the inverse operation!

The inverse operation of addition is subtraction. So, to get rid of the t on the right side, we need to subtract t from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep the equation balanced.

(3/8)t - 5 - t = t + 5 - t

Now, let's simplify. On the right side, t - t cancels out, leaving us with just 5. On the left side, we have (3/8)t - t. To subtract these terms, we need to think of t as (8/8)t (since 8/8 = 1). So we have:

(3/8)t - (8/8)t - 5 = 5

Now we can subtract the coefficients: (3/8) - (8/8) = -5/8

So, our equation becomes:

(-5/8)t - 5 = 5

We're making excellent progress, guys! We've successfully moved all the t terms to the left side of the equation. Now, we just need to isolate t completely. We're one step closer to the finish line!

Step 3: Isolating the Variable Term

Okay, we've got (-5/8)t - 5 = 5. The next step in our quest to solve for t is to isolate the term with t in it. That means we need to get rid of the -5 on the left side of the equation. How do we do that? You guessed it – by using the inverse operation!

Since we're subtracting 5, the inverse operation is addition. So, we'll add 5 to both sides of the equation:

(-5/8)t - 5 + 5 = 5 + 5

On the left side, -5 + 5 cancels out, leaving us with just (-5/8)t. On the right side, 5 + 5 = 10. So, our equation becomes:

(-5/8)t = 10

Look at that! We're almost there! The term with t is now isolated on the left side. We just have one more step to go before we have our solution. You're doing great – keep up the awesome work!

Step 4: Solving for t

We've reached the final hurdle! We're currently sitting with the equation (-5/8)t = 10. To finally solve for t, we need to get rid of the coefficient (-5/8) that's multiplying t. How do we do that? By multiplying both sides of the equation by the reciprocal of -5/8.

The reciprocal of a fraction is simply the fraction flipped upside down. So, the reciprocal of -5/8 is -8/5. Remember, multiplying a fraction by its reciprocal results in 1, which is exactly what we want!

Let's multiply both sides of the equation by -8/5:

(-8/5) * (-5/8)t = 10 * (-8/5)

On the left side, (-8/5) * (-5/8) equals 1, so we're left with just t. On the right side, we have 10 * (-8/5). To multiply a whole number by a fraction, we can think of the whole number as a fraction with a denominator of 1. So, 10 is the same as 10/1. Now we can multiply the numerators and the denominators:

(10/1) * (-8/5) = (10 * -8) / (1 * 5) = -80 / 5

Now we simplify the fraction: -80 / 5 = -16

So, our final equation is:

t = -16

We did it! We solved for t! The solution to the equation (1/8)t - 5 + (1/4)t = t + 5 is t = -16. Give yourselves a huge pat on the back, guys! You've successfully navigated the world of algebraic equations and emerged victorious.

Conclusion: You've Conquered the Equation!

Solving for t in the equation (1/8)t - 5 + (1/4)t = t + 5 might have seemed daunting at first, but by breaking it down step-by-step, we were able to conquer it! We combined like terms, moved variable terms to one side, isolated the variable term, and finally, solved for t. The key takeaway here is that complex problems can be solved by breaking them down into smaller, more manageable steps.

Remember, guys, practice makes perfect! The more you work with algebraic equations, the more comfortable you'll become with the process. So, keep practicing, keep exploring, and keep pushing your mathematical boundaries. You've got this! And the next time you encounter an equation that seems tricky, remember the steps we've covered today, and you'll be well on your way to solving it. Happy equation-solving!