Problem 21 - Beginning and Intermediate Algebra

Question

Solve each formula for the specified variable. (Leave \(\pm\) in the answers as needed.) See Examples I and 2. \(p=\sqrt{\frac{k \ell}{g}}\) for \(\ell\)

Step-by-Step Solution

Answer

\(\ell = \frac{g p^2}{k}\)

Step 1: Square Both Sides

Start by squaring both sides of the equation to eliminate the square root. The equation becomes:d(\frac{k \, \ell}{g})which simplifies to: \( p^2 = \frac{k \, \ell}{g} \)

Step 2: Isolate \(k \, \ell\)

Multiply both sides of the equation by \(g\) to isolate \(k \ell\): \( g p^2 = k \, \ell\)

Step 3: Solve for \(\ell\)

Divide both sides of the equation by \(k\) to solve for \(\ell\): \( \ell = \frac{g p^2}{k} \)

Key Concepts

Isolating Variables

When solving equations, one of the key steps is isolating the variable of interest. This means we want to get the variable by itself on one side of the equation. Here are some basic steps to follow:

Start by identifying the variable you need to solve for. In our example, we are solving for \(\ell\).

Perform operations to both sides of the equation equally. This keeps the equation balanced.

Move any other terms or coefficients to the other side. This involves adding, subtracting, multiplying, or dividing until the variable is alone.

This process of isolating variables is fundamental in algebra, helping to make complex equations more manageable.

Algebraic Manipulation

Algebraic manipulation involves using algebraic techniques to simplify or rearrange equations. This is commonly needed when isolating variables or solving equations. Here are some tools you can use:

• Multiplying or dividing both sides of an equation
• Adding or subtracting terms
• Using the distributive property to expand or factor expressions

In our example, we use several manipulations:

• Squaring both sides to get rid of the square root: \( p^2 = \frac{k \, \ell}{g} \)
• Multiplying both sides by \( g \) to isolate \( k \, \ell \: g p^2 = k \ , \ell \)
• Dividing both sides by \( k \: \ \ell = \frac{g p^2}{k} \)

By mastering these manipulative techniques, you can solve any number of algebraic equations efficiently.

Square Root Property

The square root property is a useful concept for solving equations involving square roots. Typically, to eliminate a square root, we square both sides of the equation.

In our exercise:

• The original equation is \( p = \frac{\sqrt{k \, \ell}}{g} \)
• We square both sides to remove the square root: \ p^2 = \frac{k \, \ell}{g} \

After this step, it becomes simpler algebraic manipulation.

Remember, whenever you apply the square root property:

• Check if squaring both sides introduces any extraneous solutions that don't satisfy the original equation.

Mastering the square root property will significantly help in solving complex algebraic equations involving roots.