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Chapter 1: Problem 114

Solve \(\frac{1}{a}+\frac{1}{b}=\frac{1}{x}\) for \(x\)

Short Answer

Expert verified
x = \frac{ab}{b + a}

Step by step solution

01

Find a common denominator for the left-hand side

To combine the fractions on the left-hand side, find a common denominator. The common denominator for \(\frac{1}{a}\) and \(\frac{1}{b}\) is \(ab\). Therefore, the left-hand side becomes \( \frac{b}{ab} + \frac{a}{ab} = \frac{b+a}{ab} \)
02

Rewrite the equation with the common denominator

Now rewrite the equation as: \(\frac{b + a}{ab} = \frac{1}{x}\)
03

Cross-multiply to solve for x

To solve for \(x\), cross-multiply the equation \( \frac{b + a}{ab} = \frac{1}{x} \), which gives \(x(b + a) = ab\)
04

Isolate x on one side

Divide both sides of the equation \(x(b + a) = ab\) by \(b + a\) to isolate \(x\): \(x = \frac{ab}{b + a}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominator
When working with rational equations, like \(\frac{1}{a} + \frac{1}{b} = \frac{1}{x}\), a common denominator helps combine fractions into a single expression. For example, with fractions \(\frac{1}{a}\) and \(\frac{1}{b}\), the common denominator is the product of the two denominators, \({ab}\). This lets us rewrite:
\(\frac{1}{a} + \frac{1}{b} = \frac{b}{ab} + \frac{a}{ab} = \frac{a+b}{ab}\).
Now it's easier to solve the equation. Remember, finding a common denominator simplifies combining fractions and makes the equation solvable.
Cross-Multiplication
A handy trick for solving rational equations is cross-multiplying. If you reach a stage like \(\frac{b + a}{ab} = \frac{1}{x}\), you can cross-multiply. This means multiplying the numerator of one fraction by the denominator of the other.
So, \((b + a) \times x = 1 \times ab\), making it easier to isolate the variable later. Cross-multiplication is useful because it eliminates fractions, making the equation simpler.
Isolate Variable
Once cross-multiplication simplifies the equation, you need to isolate the variable. In our example, after cross-multiplying, we get \(x(b + a) = ab\). To isolate \(x\), divide both sides by \(b + a\):
\(x = \frac{ab}{b + a}\).
Isolating the variable typically involves inverse operations like division or subtraction, allowing you to solve for the variable efficiently.

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