Solving For T: A Step-by-Step Guide To 14.2t - 25.2 = 3.8t + 26.8

Oct 10, 2025 by ADMIN 66 views

Alright, let's dive into solving this equation for 't'! Equations like this pop up all the time in math, physics, and even everyday problem-solving. So, understanding how to tackle them is super useful. We’re going to break down each step, so you can follow along easily. No stress, we’ll get through this together! First, let's rewrite the equation so we're all on the same page:

$14.2t - 25.2 = 3.8t + 26.8$

Step 1: Gather Like Terms

The main goal here is to get all the terms with 't' on one side of the equation and all the constant terms (the numbers without 't') on the other side. This makes the equation easier to manage and solve. To do this, we'll subtract $3.8t$ from both sides of the equation. This keeps the equation balanced, which is super important.

$14.2t - 3.8t - 25.2 = 3.8t - 3.8t + 26.8$

Simplifying this gives us:

$10.4t - 25.2 = 26.8$

Now, we want to isolate the term with 't' even further. We'll add $25.2$ to both sides to get rid of the constant term on the left side:

$10.4t - 25.2 + 25.2 = 26.8 + 25.2$

Which simplifies to:

$10.4t = 52$

Step 2: Isolate 't'

Now that we have all the 't' terms on one side and the constants on the other, we need to get 't' all by itself. Currently, 't' is being multiplied by $10.4$ . To undo this multiplication, we'll divide both sides of the equation by $10.4$ :

$\frac{10.4t}{10.4} = \frac{52}{10.4}$

This simplifies to:

$t = 5$

And there you have it! We've solved for 't'.

Step 3: Verification (Always a Good Idea!)

To make sure we didn't make any mistakes along the way, it’s always a good idea to plug our solution back into the original equation. This helps confirm that our answer is correct. So, let's substitute $t = 5$ into the original equation:

$14.2(5) - 25.2 = 3.8(5) + 26.8$

Now, we'll calculate each side separately:

Left side:

$14.2 * 5 = 71$

$71 - 25.2 = 45.8$

Right side:

$3.8 * 5 = 19$

$19 + 26.8 = 45.8$

Since both sides of the equation equal $45.8$ , our solution $t = 5$ is correct! Woo-hoo!

Key Concepts Used

Combining Like Terms: We grouped terms with 't' together and constant terms together to simplify the equation.
Maintaining Balance: Whatever operation we performed on one side of the equation, we also did on the other side to keep the equation balanced.
Inverse Operations: We used inverse operations (addition/subtraction and multiplication/division) to isolate 't'.
Verification: We plugged our solution back into the original equation to check our work.

Common Mistakes to Avoid

Forgetting to Distribute: If there are parentheses in the equation, make sure to distribute any multiplication over all terms inside the parentheses.
Not Maintaining Balance: Always perform the same operation on both sides of the equation to keep it balanced. This is crucial for finding the correct solution.
Sign Errors: Be careful with negative signs, especially when adding or subtracting terms. A simple sign error can throw off your entire solution.
Incorrect Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS) when simplifying expressions.

Practice Problems

Now that you've seen how to solve this equation, here are a few practice problems to test your skills:

$5.5t + 12.3 = 2.5t - 6.9$
$8.1t - 15.7 = 1.1t + 20.3$
$12.6t + 9.4 = 6.6t - 18.2$

Try solving these on your own, and then check your answers by plugging them back into the original equations. Remember, practice makes perfect!

Real-World Applications

Solving linear equations like this isn't just a math exercise; it has plenty of real-world applications. For instance:

Calculating Costs: Imagine you're comparing two phone plans. Plan A charges a monthly fee of $20 plus $0.10 per minute, while Plan B charges a monthly fee of $30 plus $0.05 per minute. You can set up an equation to find out how many minutes you need to use each month for the two plans to cost the same.
Mixing Solutions: In chemistry, you might need to mix two solutions with different concentrations to get a desired concentration. Linear equations can help you determine how much of each solution to use.
Distance, Rate, and Time: If you know the distance and rate of travel, you can use a linear equation to find the time it takes to reach your destination.

Tips for Success

Stay Organized: Keep your work neat and organized, especially when dealing with multiple steps. This will help you avoid mistakes and make it easier to check your work.
Show Your Work: Even if you can do some of the steps in your head, it's always a good idea to write them down. This will help you catch any errors and make it easier to follow your thought process.
Check Your Answers: Always plug your solution back into the original equation to make sure it's correct. This is the best way to catch any mistakes and ensure you get the right answer.
Practice Regularly: The more you practice solving linear equations, the easier it will become. Set aside some time each day or week to work on practice problems.
Seek Help When Needed: If you're struggling with a particular concept or problem, don't hesitate to ask for help. Talk to your teacher, a tutor, or a classmate. There are also plenty of online resources available, such as videos and tutorials.

Conclusion

So, we successfully solved the equation $14.2t - 25.2 = 3.8t + 26.8$ and found that $t = 5$ . We walked through each step, highlighting key concepts, common mistakes to avoid, and real-world applications. With practice and a solid understanding of these concepts, you'll be solving linear equations like a pro in no time! Keep up the great work, and don't forget to have fun with it!