Problem 100
Question
Solve each equation for the indicated variable. Assume all denominators are nonzero. $$\frac{1}{R}=\frac{1}{r_{1}}+\frac{1}{r_{2}}, \quad \text { for } R$$
Step-by-Step Solution
Verified Answer
The solution for \(R\) is \(\frac{r_1 \times r_2}{r_1 + r_2}\).
1Step 1 - Identify the equation
The given equation is \(\frac{1}{R} = \frac{1}{r_{1}} + \frac{1}{r_{2}}\). We need to solve this equation for the variable \(R\).
2Step 2 - Find a common denominator
Combine the fractions on the right side of the equation by finding a common denominator. This gives us \(\frac{1}{R} = \frac{r_{1} + r_{2}}{r_{1} \times r_{2}}\).
3Step 3 - Invert both sides of the equation
To isolate \(R\), invert both sides of the equation. This results in \(R = \frac{r_{1} \times r_{2}}{r_{1} + r_{2}}\).
Key Concepts
common denominatorvariable isolationfractional equations
common denominator
When solving equations with fractions, a common denominator is your best friend. It's a shared multiple of the denominators in the fraction. By finding one, you can combine fractions with ease.
Let’s take our equation as an example: \ \ \(\frac{1}{R} = \frac{1}{r_{1}} + \frac{1}{r_{2}} \). The fractions \(\frac{1}{r_{1}}\) and \(\frac{1}{r_{2}}\) have different denominators but we want to add them. The common denominator for \(\frac{1}{r_{1}}\) and \(\frac{1}{r_{2}}\) is \(r_{1} \times r_{2}\).
Combining the fractions then gives us:
\(\frac{1}{r_{1}} + \frac{1}{r_{2}} = \frac{r_{2}}{r_{1} \times r_{2}} + \frac{r_{1}}{r_{1} \times r_{2}} \). By adding the numerators, we achieve:
\(\frac{r_{1} + r_{2}}{r_{1} \times r_{2}} \).
This is how using a common denominator helps us to combine the fractions into one single fraction.
Let’s take our equation as an example: \ \ \(\frac{1}{R} = \frac{1}{r_{1}} + \frac{1}{r_{2}} \). The fractions \(\frac{1}{r_{1}}\) and \(\frac{1}{r_{2}}\) have different denominators but we want to add them. The common denominator for \(\frac{1}{r_{1}}\) and \(\frac{1}{r_{2}}\) is \(r_{1} \times r_{2}\).
Combining the fractions then gives us:
\(\frac{1}{r_{1}} + \frac{1}{r_{2}} = \frac{r_{2}}{r_{1} \times r_{2}} + \frac{r_{1}}{r_{1} \times r_{2}} \). By adding the numerators, we achieve:
\(\frac{r_{1} + r_{2}}{r_{1} \times r_{2}} \).
This is how using a common denominator helps us to combine the fractions into one single fraction.
variable isolation
Isolating the variable means you get the variable on one side of the equation and everything else on the other.
For example, in the equation \(\frac{1}{R} = \frac{r_{1} + r_{2}}{r_{1} \times r_{2}}\), our goal is to solve for \(R\).
To isolate \(R\), we need to get rid of the fraction. Fractions can be tricky but there are strategies to make this simpler.
Step: Invert both sides of the equation. This flips the numerators and denominators, resulting in:
\(\frac{1}{\frac{1}{R}} = \frac{r_{1} \times r_{2}}{r_{1} + r_{2}}\).
The left side becomes simply \(R\), simplifying the equation significantly to:
\(R = \frac{r_{1} \times r_{2}}{r_{1} + r_{2}}\).
Now, \(R\) is isolated and we have our solution.
For example, in the equation \(\frac{1}{R} = \frac{r_{1} + r_{2}}{r_{1} \times r_{2}}\), our goal is to solve for \(R\).
To isolate \(R\), we need to get rid of the fraction. Fractions can be tricky but there are strategies to make this simpler.
Step: Invert both sides of the equation. This flips the numerators and denominators, resulting in:
\(\frac{1}{\frac{1}{R}} = \frac{r_{1} \times r_{2}}{r_{1} + r_{2}}\).
The left side becomes simply \(R\), simplifying the equation significantly to:
\(R = \frac{r_{1} \times r_{2}}{r_{1} + r_{2}}\).
Now, \(R\) is isolated and we have our solution.
fractional equations
Handling fractional equations can seem daunting, but the steps become more manageable when broken down.
In our initial example, \(\frac{1}{R} = \frac{1}{r_{1}} + \frac{1}{r_{2}}\), the goal is to isolate \(R\).
The first step is to combine the fractions on one side. By finding a common denominator as previously discussed, we add the fractions and get:
\(\frac{1}{R} = \frac{r_{1} + r_{2}}{r_{1} \times r_{2}}\).
Next, we isolate the variable by inverting both sides of the equation. This means we reciprocate or flip the fractions, turning the equation into:
\(\frac{1}{\frac{1}{R}} = \frac{r_{1} \times r_{2}}{r_{1} + r_{2}}\).
The left side simplifies directly to \(R\), leaving us with:
\(R = \frac{r_{1} \times r_{2}}{r_{1} + r_{2}}\).
By methodically combining and simplifying fractions, solving fractional equations becomes a straightforward process.
In our initial example, \(\frac{1}{R} = \frac{1}{r_{1}} + \frac{1}{r_{2}}\), the goal is to isolate \(R\).
The first step is to combine the fractions on one side. By finding a common denominator as previously discussed, we add the fractions and get:
\(\frac{1}{R} = \frac{r_{1} + r_{2}}{r_{1} \times r_{2}}\).
Next, we isolate the variable by inverting both sides of the equation. This means we reciprocate or flip the fractions, turning the equation into:
\(\frac{1}{\frac{1}{R}} = \frac{r_{1} \times r_{2}}{r_{1} + r_{2}}\).
The left side simplifies directly to \(R\), leaving us with:
\(R = \frac{r_{1} \times r_{2}}{r_{1} + r_{2}}\).
By methodically combining and simplifying fractions, solving fractional equations becomes a straightforward process.
Other exercises in this chapter
Problem 98
Solve each equation for the indicated variable. Assume all denominators are nonzero. $$x^{2 / 3}+y^{2 / 3}=a^{2 / 3}, \quad \text { for } y$$
View solution Problem 98
Find the values of \(a, b,\) and \(c\) for which the quadratic equation $$ a x^{2}+b x+c=0 $$ has the given numbers as solutions. (Hint: Use the zero-factor pro
View solution Problem 101
Show that \(-2+i\) is a solution of the equation \(x^{2}+4 x+5=0\)
View solution Problem 101
Solve each equation for the indicated variable. Assume all denominators are nonzero. $$\frac{E}{e}=\frac{R+r}{r}, \quad \text { for } e$$
View solution