Solving For R How To Make R The Subject Of The Formula 1/p=b/t+c/r

In the realm of mathematical manipulations, rearranging formulas to isolate a specific variable is a fundamental skill. It empowers us to solve for unknowns, analyze relationships between variables, and gain deeper insights into the underlying principles governing equations. This article delves into the process of making $r$ the subject of the relation $\frac{1}{p} = \frac{b}{t} + \frac{c}{r}$, providing a comprehensive guide with step-by-step instructions and explanations.

Understanding the Equation

Before we embark on the transformation journey, let's first grasp the essence of the equation $\frac{1}{p} = \frac{b}{t} + \frac{c}{r}$. This equation represents a relationship between several variables: $p$, $b$, $t$, $c$, and $r$. Our mission is to isolate $r$ on one side of the equation, expressing it in terms of the other variables.

The equation involves fractions, which adds a layer of complexity. To effectively manipulate the equation, we'll employ techniques such as finding a common denominator, cross-multiplication, and algebraic rearrangement. By mastering these techniques, you'll be well-equipped to tackle similar equation-solving challenges.

Step-by-Step Solution: Isolating $r$

Now, let's embark on the step-by-step journey of making $r$ the subject of the formula. We'll break down the process into manageable steps, providing explanations and justifications along the way.

Step 1: Isolate the term containing $r$

The initial step is to isolate the term that contains the variable $r$, which is $\frac{c}{r}$. To achieve this, we subtract $\frac{b}{t}$ from both sides of the equation:

$\frac{1}{p} - \frac{b}{t} = \frac{c}{r}$

This step effectively moves the term $\frac{b}{t}$ to the left-hand side, leaving the term containing $r$ isolated on the right-hand side.

Step 2: Find a common denominator on the left-hand side

To combine the fractions on the left-hand side, we need to find a common denominator. The least common multiple of $p$ and $t$ is $pt$. We rewrite the fractions with the common denominator:

$\frac{t}{pt} - \frac{bp}{pt} = \frac{c}{r}$

Step 3: Combine the fractions on the left-hand side

Now that the fractions on the left-hand side have a common denominator, we can combine them:

$\frac{t - bp}{pt} = \frac{c}{r}$

This step simplifies the left-hand side into a single fraction, making the equation more manageable.

Step 4: Cross-multiplication

To eliminate the fractions, we employ cross-multiplication. This involves multiplying the numerator of the left-hand side by the denominator of the right-hand side, and vice versa:

$r(t - bp) = c(pt)$

Cross-multiplication effectively removes the fractions, transforming the equation into a more linear form.

Step 5: Isolate $r$

The final step is to isolate $r$ by dividing both sides of the equation by $(t - bp)$:

$r = \frac{cpt}{t - bp}$

This step isolates $r$ on the left-hand side, expressing it in terms of the other variables. We have successfully made $r$ the subject of the formula.

The Final Result

After meticulous manipulation, we have successfully made $r$ the subject of the relation. The final result is:

$r = \frac{cpt}{t - bp}$

This equation expresses $r$ in terms of $c$, $p$, $t$, and $b$. It allows us to calculate the value of $r$ given the values of the other variables. Moreover, it reveals the relationship between $r$ and the other variables, providing valuable insights into the equation.

Applications and Significance

The ability to rearrange formulas and isolate variables has far-reaching applications across various fields. In physics, it allows us to solve for unknown quantities in equations of motion and energy. In engineering, it enables us to design structures and systems by manipulating equations that govern their behavior. In economics, it helps us analyze economic models and predict market trends. The skill of making a variable the subject of a formula is a cornerstone of problem-solving in mathematics and its applications.

Furthermore, understanding how to manipulate equations enhances our analytical thinking and problem-solving abilities. It fosters a deeper comprehension of the relationships between variables and empowers us to make informed decisions based on mathematical models.

Practice Problems

To solidify your understanding and hone your skills, let's tackle a couple of practice problems:

1. Solve for $x$ in the equation: $\frac{1}{a} = \frac{2}{x} + \frac{3}{b}$
2. Make $v$ the subject of the formula: $K = \frac{1}{2}mv^2$

By working through these problems, you'll reinforce the techniques we've discussed and gain confidence in your ability to rearrange formulas.

Conclusion

In this comprehensive guide, we've explored the process of making $r$ the subject of the relation $\frac{1}{p} = \frac{b}{t} + \frac{c}{r}$. We've broken down the solution into manageable steps, providing explanations and justifications along the way. By mastering the techniques discussed in this article, you'll be well-equipped to tackle similar equation-solving challenges and unlock the power of mathematical manipulation.

Remember, the ability to rearrange formulas is a valuable skill with applications across various fields. It empowers us to solve for unknowns, analyze relationships between variables, and gain deeper insights into the world around us. So, embrace the challenge, practice diligently, and unlock the potential of mathematical manipulation.

Keep practicing and exploring the fascinating world of mathematics! With dedication and perseverance, you'll become a master of equation manipulation and unlock new realms of mathematical understanding.