Solve The Given Equation. Check Your Solution.$\[ \frac{t^2}{t+3} + \frac{3t}{t+3} = 1 \\]Find \[$t\$\].

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Introduction

In this article, we will delve into solving a given equation, which involves combining fractions and simplifying the resulting expression. The equation to be solved is t2t+3+3tt+3=1\frac{t^2}{t+3} + \frac{3t}{t+3} = 1. Our goal is to find the value of tt that satisfies this equation.

Step 1: Combine the Fractions

The first step in solving this equation is to combine the two fractions on the left-hand side. Since both fractions have the same denominator, t+3t+3, we can add them together by adding the numerators and keeping the common denominator.

t2t+3+3tt+3=t2+3tt+3\frac{t^2}{t+3} + \frac{3t}{t+3} = \frac{t^2 + 3t}{t+3}

Step 2: Simplify the Expression

Now that we have combined the fractions, we can simplify the resulting expression. To do this, we can factor out a common term from the numerator.

t2+3tt+3=t(t+3)t+3\frac{t^2 + 3t}{t+3} = \frac{t(t+3)}{t+3}

Step 3: Cancel Out the Common Term

Since the numerator and denominator have a common term, t+3t+3, we can cancel it out. This will leave us with a simplified expression.

t(t+3)t+3=t\frac{t(t+3)}{t+3} = t

Step 4: Equate the Simplified Expression to 1

Now that we have simplified the expression, we can equate it to 1, as given in the original equation.

t=1t = 1

Conclusion

In this article, we have solved the given equation by combining fractions, simplifying the resulting expression, and canceling out the common term. The final solution is t=1t = 1. This means that the value of tt that satisfies the equation is 1.

Why is this Solution Valid?

To verify that this solution is valid, we can substitute t=1t = 1 back into the original equation and check if it holds true.

121+3+3(1)1+3=14+34=1\frac{1^2}{1+3} + \frac{3(1)}{1+3} = \frac{1}{4} + \frac{3}{4} = 1

As we can see, the equation holds true when t=1t = 1. Therefore, this solution is valid.

What if the Solution is Not Valid?

If the solution is not valid, it means that the equation has no solution or that the solution is extraneous. In this case, we would need to re-examine the steps we took to solve the equation and check for any errors.

Tips and Tricks

When solving equations involving fractions, it's essential to combine the fractions and simplify the resulting expression. This will help you avoid errors and make it easier to find the solution.

Common Mistakes to Avoid

When solving equations involving fractions, some common mistakes to avoid include:

  • Not combining the fractions
  • Not simplifying the resulting expression
  • Canceling out the wrong term

By avoiding these common mistakes, you can ensure that your solution is valid and accurate.

Real-World Applications

Solving equations involving fractions has many real-world applications, such as:

  • Physics: When solving problems involving motion, you may encounter equations involving fractions.
  • Engineering: When designing systems, you may need to solve equations involving fractions to determine the optimal solution.
  • Economics: When analyzing economic data, you may need to solve equations involving fractions to determine the relationship between variables.

Conclusion

Introduction

In our previous article, we solved the given equation t2t+3+3tt+3=1\frac{t^2}{t+3} + \frac{3t}{t+3} = 1 and found that the value of tt that satisfies this equation is t=1t = 1. In this article, we will answer some frequently asked questions related to solving this equation.

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to combine the two fractions on the left-hand side. Since both fractions have the same denominator, t+3t+3, we can add them together by adding the numerators and keeping the common denominator.

Q: Why do we need to simplify the expression?

A: We need to simplify the expression to make it easier to solve the equation. By simplifying the expression, we can cancel out any common terms and make the equation more manageable.

Q: What is the purpose of canceling out the common term?

A: The purpose of canceling out the common term is to simplify the expression and make it easier to solve the equation. By canceling out the common term, we can eliminate any unnecessary variables and make the equation more straightforward.

Q: How do we know if the solution is valid?

A: To verify that the solution is valid, we can substitute the value of tt back into the original equation and check if it holds true. If the equation holds true, then the solution is valid.

Q: What if the solution is not valid?

A: If the solution is not valid, it means that the equation has no solution or that the solution is extraneous. In this case, we would need to re-examine the steps we took to solve the equation and check for any errors.

Q: What are some common mistakes to avoid when solving equations involving fractions?

A: Some common mistakes to avoid when solving equations involving fractions include:

  • Not combining the fractions
  • Not simplifying the resulting expression
  • Canceling out the wrong term

Q: How does solving equations involving fractions apply to real-world problems?

A: Solving equations involving fractions has many real-world applications, such as:

  • Physics: When solving problems involving motion, you may encounter equations involving fractions.
  • Engineering: When designing systems, you may need to solve equations involving fractions to determine the optimal solution.
  • Economics: When analyzing economic data, you may need to solve equations involving fractions to determine the relationship between variables.

Q: What are some tips and tricks for solving equations involving fractions?

A: Some tips and tricks for solving equations involving fractions include:

  • Combining the fractions as soon as possible
  • Simplifying the resulting expression
  • Canceling out any common terms
  • Verifying the solution by substituting the value of tt back into the original equation

Conclusion

In conclusion, solving the given equation involves combining fractions, simplifying the resulting expression, and canceling out the common term. By following the steps outlined in this article, you can solve equations involving fractions and apply the concepts to real-world problems. Remember to avoid common mistakes and verify the solution to ensure that it is valid.

Frequently Asked Questions

  • Q: What is the first step in solving the equation? A: The first step in solving the equation is to combine the two fractions on the left-hand side.
  • Q: Why do we need to simplify the expression? A: We need to simplify the expression to make it easier to solve the equation.
  • Q: What is the purpose of canceling out the common term? A: The purpose of canceling out the common term is to simplify the expression and make it easier to solve the equation.
  • Q: How do we know if the solution is valid? A: To verify that the solution is valid, we can substitute the value of tt back into the original equation and check if it holds true.

Additional Resources

  • For more information on solving equations involving fractions, please refer to our previous article.
  • For additional practice problems, please see the exercises at the end of this article.
  • For more resources on math and science, please visit our website.