Problem 74 - Precalculus Student Solutions Manual 5th

Question

Solve each equation for the indicated variable. Assume no denominators are \(0 .\) $$E=\frac{e^{2} k}{2 r}, \quad \text { for } e$$

Step-by-Step Solution

Answer

e = \(\sqrt{\frac{2rE}{k}}\)

Step 1: Isolate the variable

First, multiply both sides of the equation by the denominator to eliminate it. Multiply both sides by \(2r\): \[2rE = e^2 k\]

Step 2: Solve for e^2

Next, isolate \(e^2\) by dividing both sides of the equation by \(k\): \[e^2 = \frac{2rE}{k}\]

Step 3: Take the square root

Finally, solve for \(e\) by taking the square root of both sides of the equation: \[e = \sqrt{\frac{2rE}{k}}\]

Key Concepts

Variable Isolation

Variable isolation is an essential concept in algebra. It involves rearranging an equation to get the unknown variable by itself on one side of the equation. This makes it easier to solve for that variable.
For example, in our problem, we are given the equation: \(E=\frac{e^{2} k}{2 r}\) We want to solve for \(e\). To do this, we need to isolate \(e\) by performing operations that cancel out all other terms on the same side of the equation. This usually involves steps like addition, subtraction, multiplication, or division. By following these operations systematically, we can get the variable by itself and find its value.

Eliminating Denominators

Eliminating denominators is a crucial step when dealing with equations involving fractions. It simplifies the equation and makes solving it much easier.
In our equation: \(E=\frac{e^{2} k}{2 r}\) The denominator is \(2r\). To eliminate this denominator, we multiply both sides of the equation by \(2r\). This gives us: \(2rE = e^2 k\) By removing the fraction, the equation becomes simpler and more straightforward to solve. Now, we can focus on isolating the variable \(e\).

Square Root

Taking the square root is a common method used when solving equations that involve squares of a variable. It helps in finding the value of the variable that, when squared, gives the original number.
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