Introduction to Solving Equations with Variables in the Denominator

Oct 13, 2025 by miodeya.com 92 views

Introduction to Solving Equations with Variables in the Denominator

When solving equations that involve variables in the denominator, it's essential to follow a step-by-step approach to ensure accuracy and avoid common pitfalls. In this article, we will focus on solving the equation $\frac{15}{4t^2} = \frac{2}{3}$ for the variable $t$ . This equation involves fractions and a quadratic expression in the denominator, making it a challenging problem that requires careful manipulation.

Step 1: Multiply Both Sides by the Least Common Multiple (LCM) of the Denominators

To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of $4t^2$ and $3$ is $12t^2$ . Multiplying both sides by $12t^2$ will eliminate the fractions and allow us to work with a simpler equation.

\frac{15}{4t^2} \cdot 12t^2 = \frac{2}{3} \cdot 12t^2

Step 2: Simplify the Equation

After multiplying both sides by $12t^2$ , we can simplify the equation by canceling out the common factors.

\frac{15 \cdot 12}{4} = \frac{2 \cdot 12t^2}{3}

Step 3: Simplify the Left Side of the Equation

We can simplify the left side of the equation by canceling out the common factors.

\frac{180}{4} = \frac{2 \cdot 12t^2}{3}

Step 4: Simplify the Right Side of the Equation

We can simplify the right side of the equation by canceling out the common factors.

45 = \frac{2 \cdot 12t^2}{3}

Step 5: Multiply Both Sides by 3 to Eliminate the Fraction

To eliminate the fraction on the right side of the equation, we can multiply both sides by 3.

45 \cdot 3 = 2 \cdot 12t^2

Step 6: Simplify the Equation

After multiplying both sides by 3, we can simplify the equation by canceling out the common factors.

135 = 24t^2

Step 7: Divide Both Sides by 24 to Isolate $t^2$

To isolate $t^2$ , we can divide both sides of the equation by 24.

\frac{135}{24} = t^2

Step 8: Simplify the Left Side of the Equation

We can simplify the left side of the equation by canceling out the common factors.

\frac{45}{8} = t^2

Step 9: Take the Square Root of Both Sides to Solve for $t$

To solve for $t$ , we can take the square root of both sides of the equation.

\sqrt{\frac{45}{8}} = \sqrt{t^2}

Step 10: Simplify the Left Side of the Equation

We can simplify the left side of the equation by canceling out the common factors.

\frac{\sqrt{45}}{\sqrt{8}} = t

Step 11: Simplify the Left Side of the Equation

We can simplify the left side of the equation by canceling out the common factors.

\frac{\sqrt{9 \cdot 5}}{\sqrt{4 \cdot 2}} = t

Step 12: Simplify the Left Side of the Equation

We can simplify the left side of the equation by canceling out the common factors.

\frac{3\sqrt{5}}{2\sqrt{2}} = t

Step 13: Rationalize the Denominator

To rationalize the denominator, we can multiply both the numerator and denominator by $\sqrt{2}$ .

\frac{3\sqrt{5} \cdot \sqrt{2}}{2\sqrt{2} \cdot \sqrt{2}} = t

Step 14: Simplify the Left Side of the Equation

We can simplify the left side of the equation by canceling out the common factors.

\frac{3\sqrt{10}}{4} = t

The final answer is $\boxed{\frac{3\sqrt{10}}{4}}$ .

Introduction

Solving equations with variables in the denominator can be a challenging task, but with the right approach, it can be done with ease. In this article, we will provide a Q&A section to help you better understand the concept and provide additional insights.

Q: What is the least common multiple (LCM) of the denominators?

A: The least common multiple (LCM) of the denominators is the smallest number that both denominators can divide into evenly. In the case of the equation $\frac{15}{4t^2} = \frac{2}{3}$ , the LCM of $4t^2$ and $3$ is $12t^2$ .

Q: Why do we need to multiply both sides of the equation by the LCM?

A: We need to multiply both sides of the equation by the LCM to eliminate the fractions and make it easier to work with the equation.

Q: What is the next step after multiplying both sides by the LCM?

A: After multiplying both sides by the LCM, we need to simplify the equation by canceling out the common factors.

Q: How do we simplify the equation?

A: We simplify the equation by canceling out the common factors on both sides of the equation.

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

A: The final step in solving the equation is to take the square root of both sides to solve for $t$ .

Q: What is the final answer?

A: The final answer is $\boxed{\frac{3\sqrt{10}}{4}}$ .

Q: What are some common mistakes to avoid when solving equations with variables in the denominator?

A: Some common mistakes to avoid when solving equations with variables in the denominator include:

Not multiplying both sides of the equation by the LCM
Not simplifying the equation by canceling out the common factors
Not taking the square root of both sides to solve for $t$

Q: How can I practice solving equations with variables in the denominator?

A: You can practice solving equations with variables in the denominator by working through examples and exercises. You can also try solving equations with different denominators and variables to see how the process works.

Q: What are some real-world applications of solving equations with variables in the denominator?

A: Solving equations with variables in the denominator has many real-world applications, including:

Physics: Solving equations with variables in the denominator is used to calculate distances, velocities, and accelerations.
Engineering: Solving equations with variables in the denominator is used to design and optimize systems.
Economics: Solving equations with variables in the denominator is used to model and analyze economic systems.

Q: Can I use a calculator to solve equations with variables in the denominator?

A: Yes, you can use a calculator to solve equations with variables in the denominator. However, it's always a good idea to check your work by hand to make sure you understand the process.

Q: What are some tips for solving equations with variables in the denominator?

A: Some tips for solving equations with variables in the denominator include:

Make sure to multiply both sides of the equation by the LCM
Simplify the equation by canceling out the common factors
Take the square root of both sides to solve for $t$
Check your work by hand to make sure you understand the process

By following these tips and practicing solving equations with variables in the denominator, you'll become more confident and proficient in solving these types of equations.