Solving For T In The Equation (2t-1)/(t+3) = -2

Sam Evans · · 5 min read

Table Of Content

  1. Understanding the Problem
  2. Step-by-Step Solution
    1. 1. Eliminate the Fraction
    2. 2. Gather the t Terms
    3. 3. Isolate the t Term
    4. 4. Solve for t
  3. Checking Our Answer
    1. Simplifying the Numerator:
    2. Simplifying the Denominator:
  4. Analyzing the Answer Choices
  5. Conclusion: Mastering Algebraic Equations

Hey there, math enthusiasts! Today, we're diving into a fun little algebraic equation. We've got to figure out for what value of t the fraction $\frac{2t-1}{t+3}$ equals -2. It might sound a bit tricky at first, but trust me, we'll break it down step by step and it will be so easy.

Understanding the Problem

So, the problem presents us with an equation involving a variable, t. Our mission, should we choose to accept it (and I hope you do!), is to isolate t and find its value. This means we need to manipulate the equation using algebraic principles until we have t all by itself on one side of the equals sign. Before we jump into the solution, let's just get a lay of the land. We see a fraction on one side, which might seem intimidating, but don't worry, we've got tools to deal with that. We've also got a -2 on the other side, which is just a constant, so that's nice and straightforward. The key here is to remember the golden rule of algebra: whatever you do to one side of the equation, you've got to do to the other side. This keeps the equation balanced and ensures we get the correct answer. Think of it like a seesaw – if you add weight to one side, you've got to add the same weight to the other side to keep it level. So, with our equation $ rac{2t-1}{t+3} = -2$, we're aiming to find the specific value of t that makes this equation true. Now, let's roll up our sleeves and get started!

Step-by-Step Solution

1. Eliminate the Fraction

The first thing we want to do is get rid of that fraction. Fractions in equations can be a bit of a headache, so let's make our lives easier by multiplying both sides of the equation by the denominator, which is (t + 3). This will cancel out the denominator on the left side and leave us with a much simpler equation to work with. So, we have:

(t+3)∗2t−1t+3=−2∗(t+3)(t + 3) * \frac{2t - 1}{t + 3} = -2 * (t + 3)

On the left side, the (t + 3) in the numerator and the (t + 3) in the denominator cancel each other out, leaving us with just (2t - 1). On the right side, we need to distribute the -2 across the (t + 3), which means we multiply -2 by both t and 3. This gives us:

2t−1=−2t−62t - 1 = -2t - 6

2. Gather the t Terms

Now that we've eliminated the fraction, our equation looks much friendlier. We have t terms on both sides of the equation, so let's gather them together on one side. A common strategy is to move the t term with the smaller coefficient. In this case, we have 2t on the left and -2t on the right. Since -2t is smaller than 2t, we'll add 2t to both sides of the equation. This will cancel out the -2t on the right side and move all the t terms to the left side. Adding 2t to both sides gives us:

2t−1+2t=−2t−6+2t2t - 1 + 2t = -2t - 6 + 2t

Combining like terms, we get:

4t−1=−64t - 1 = -6

3. Isolate the t Term

We're getting closer! Now we have all the t terms on the left side, but we still have that pesky -1 hanging around. To isolate the t term, we need to get rid of the -1. We can do this by adding 1 to both sides of the equation. This will cancel out the -1 on the left side and leave us with just the 4t term. Adding 1 to both sides gives us:

4t−1+1=−6+14t - 1 + 1 = -6 + 1

Simplifying, we get:

4t=−54t = -5

4. Solve for t

Finally, we're in the home stretch! We have 4t equals -5, and we want to find the value of just t. To do this, we need to divide both sides of the equation by 4. This will isolate t on the left side and give us our answer. Dividing both sides by 4 gives us:

4t4=−54\frac{4t}{4} = \frac{-5}{4}

Simplifying, we get:

t=−54t = -\frac{5}{4}

So, there you have it! The value of t that satisfies the equation is t = -5/4. We've successfully navigated the algebraic waters and found our treasure!

Checking Our Answer

Now, before we declare victory and move on to the next challenge, it's always a good idea to check our answer. This helps us ensure we haven't made any silly mistakes along the way. To check our answer, we'll substitute our value of t back into the original equation and see if it holds true. Our original equation was:

2t−1t+3=−2\frac{2t - 1}{t + 3} = -2

We found that t = -5/4, so let's plug that in:

2(−54)−1(−54)+3=−2\frac{2(-\frac{5}{4}) - 1}{(-\frac{5}{4}) + 3} = -2

Let's simplify the numerator and the denominator separately.

Simplifying the Numerator:

2(−54)−1=−104−1=−52−1=−52−22=−722(-\frac{5}{4}) - 1 = -\frac{10}{4} - 1 = -\frac{5}{2} - 1 = -\frac{5}{2} - \frac{2}{2} = -\frac{7}{2}

Simplifying the Denominator:

(−54)+3=−54+124=74(-\frac{5}{4}) + 3 = -\frac{5}{4} + \frac{12}{4} = \frac{7}{4}

Now, let's substitute these simplified values back into our equation:

−7274=−2\frac{-\frac{7}{2}}{\frac{7}{4}} = -2

To divide fractions, we multiply by the reciprocal of the denominator:

−72∗47=−2-\frac{7}{2} * \frac{4}{7} = -2

The 7s cancel out, and we can simplify the fraction:

−12∗4=−2-\frac{1}{2} * 4 = -2

−2=−2-2 = -2

Voila! Our equation holds true. This confirms that our answer, t = -5/4, is indeed correct. We've not only solved the problem, but we've also verified our solution, which is always a good practice in mathematics. Give yourselves a pat on the back, guys! You've earned it.

Analyzing the Answer Choices

Okay, so we've diligently worked through the problem and arrived at our solution: t = -5/4. Now, let's take a look at the answer choices provided and see which one matches our result. This is a crucial step, especially in multiple-choice scenarios, as it helps us ensure we're selecting the correct option.

The answer choices are:

A. $-\frac{7}{4}$ B. $-\frac{5}{4}$ C. $-\frac{5}{12}$ D. $\frac{7}{4}$ E. There is no value of $t$ satisfying this equation.

Comparing our solution, t = -5/4, with the answer choices, we can clearly see that option B matches our result. This is fantastic news! It confirms that our step-by-step solution has led us to the correct answer. But let's not just stop there. It's always beneficial to briefly analyze the other answer choices to understand why they are incorrect. This can help us solidify our understanding of the problem and the solution process.

  • Option A (-7/4): This value is close to our correct answer, but the numerator is different. This might be a result of a small calculation error during the solving process.
  • Option C (-5/12): This value has the correct numerator but a different denominator. This could indicate an error in handling the fractions or simplifying the equation.
  • Option D (7/4): This value has the correct numbers but the wrong sign. This suggests a potential mistake in dealing with the negative signs in the equation.
  • Option E (There is no value of t satisfying this equation): We've clearly found a value of t that satisfies the equation, so this option is incorrect. This option is often a distractor, designed to catch students who might have made a mistake or are unsure of their solution.

By analyzing the incorrect answer choices, we can gain a deeper understanding of the common errors that might occur when solving this type of equation. This not only reinforces our understanding but also helps us avoid similar mistakes in the future. So, the correct answer is undoubtedly B. -5/4. We've not only solved the problem but also thoroughly analyzed the solution and the answer choices. Great job, everyone!

Conclusion: Mastering Algebraic Equations

Alright, mathletes, we've conquered another algebraic challenge! We successfully navigated the equation $\frac{2t-1}{t+3} = -2$, found the value of t, and even checked our answer to make sure we were spot on. Woo-hoo! This journey through solving for t has been more than just finding a numerical answer; it's been about honing our problem-solving skills and building confidence in our algebraic abilities. Remember, math isn't just about memorizing formulas and procedures; it's about understanding the underlying concepts and applying them strategically. In this case, we tackled a fraction, gathered like terms, isolated the variable, and verified our solution. These are fundamental techniques that will serve you well in more advanced mathematical adventures.

The key takeaway here is that algebraic equations, even those that might initially seem daunting, can be broken down into manageable steps. By following a systematic approach, like the one we used today, we can unravel the complexities and arrive at the solution. So, next time you encounter an equation that makes you scratch your head, take a deep breath, remember our adventure today, and break it down step by step. You've got this!

And remember, practice makes perfect. The more you work with algebraic equations, the more comfortable and confident you'll become. So, keep exploring, keep questioning, and keep solving. The world of mathematics is vast and fascinating, and there's always something new to discover. Until next time, keep those math muscles flexed and ready for action! You guys are awesome!

Related Posts