Textbook Question
Solve each equation. See Example 2. x/(x-1) - 1/(x+1) = 2/(x^2-1)
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1. Equations & Inequalities
Rational Equations
2:54 minutesProblem 108
Textbook Question
Solve each equation for the specified variable. (Assume all denominators are nonzero.) 1/R=1/r_1 + 1/r_2, for R
Verified step by step guidance1
Start with the given equation: \( \frac{1}{R} = \frac{1}{r_1} + \frac{1}{r_2} \).
To eliminate the fractions, find a common denominator for the right side of the equation, which is \( r_1 \times r_2 \).
Rewrite the right side with the common denominator: \( \frac{1}{R} = \frac{r_2}{r_1 \times r_2} + \frac{r_1}{r_1 \times r_2} \).
Combine the fractions on the right side: \( \frac{1}{R} = \frac{r_2 + r_1}{r_1 \times r_2} \).
Take the reciprocal of both sides to solve for \( R \): \( R = \frac{r_1 \times r_2}{r_1 + r_2} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Reciprocal Relationships
In algebra, a reciprocal relationship involves the concept of fractions where the reciprocal of a number 'x' is '1/x'. This is particularly important in equations involving rates or resistances, as seen in the equation 1/R = 1/r_1 + 1/r_2, where R represents the total resistance in a parallel circuit formed by resistances r_1 and r_2.
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Solving for a Variable
Solving for a variable means isolating that variable on one side of the equation. In the context of the given equation, we need to manipulate the equation to express R in terms of r_1 and r_2. This often involves algebraic operations such as addition, subtraction, multiplication, and division to rearrange the equation appropriately.
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Nonzero Denominators
The assumption that all denominators are nonzero is crucial in algebraic equations involving fractions. It ensures that the operations performed do not lead to undefined expressions. In the context of the equation provided, it guarantees that r_1 and r_2 cannot be zero, which would make the terms 1/r_1 and 1/r_2 undefined, thus affecting the validity of the solution.
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