Problem 72 - Intermediate Algebra

Question

Solve for the specified variable. $$ K=\frac{M v_{0}^{2}+I w^{2}}{2} \quad \text { for } I $$

Step-by-Step Solution

Answer

\( I = \frac{2K - M v_{0}^{2}}{w^{2}} \)

Step 1: Identify the Formula

We are given the formula \( K = \frac{M v_{0}^{2}+I w^{2}}{2} \). Our task is to solve for \( I \).

Step 2: Eliminate the Fraction

Multiply both sides of the equation by 2 to clear the fraction, which gives: \( 2K = M v_{0}^{2} + I w^{2} \).

Step 3: Isolate the Term Containing I

Subtract \( M v_{0}^{2} \) from both sides to isolate the term with \( I \): \( 2K - M v_{0}^{2} = I w^{2} \).

Step 4: Solve for I

Divide both sides by \( w^{2} \) to solve for \( I \): \( I = \frac{2K - M v_{0}^{2}}{w^{2}} \).

Key Concepts

Isolating Variables

Isolating a variable means getting the variable alone on one side of an equation. This is crucial for solving equations. We achieve this by reversing the operations applied to the variable, using inverse operations. In our example, we need to solve for \( I \) in the equation: \[ K = \frac{M v_{0}^{2}+I w^{2}}{2} \] To isolate \( I \), we must "move" other terms to the opposite side of the equation. This often involves negating terms, adding or subtracting terms, and performing similar operations to each side of the equation. Once we have isolated the term including \( I \), the goal is to have all other values away from \( I \) by using division or multiplication. This process may take several steps, but patience is key. Each step gets you closer to isolating the desired variable.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging, adding, subtracting, multiplying, or dividing parts of an equation to simplify it or solve for a specific variable. In our exercise, we manipulated the equation to solve for \( I \), which involved:

• Recognizing and clearing fractions by multiplying both sides of the equation.
• Moving terms involving \( I \) to one side by subtracting unwanted terms from each side.
• Finally, using division to fully isolate \( I \).

These manipulations ensure accuracy and efficiency in solving. Each rearrangement step should maintain the equation's balance by applying equal changes to both sides. This is key to keeping the equation valid and obtaining a correct solution.

Fraction Elimination

Fractions can often complicate equations, making them harder to solve. Fraction elimination involves clearing fractions to simplify the equation. This is done by multiplying every term by the denominator, thus removing the fraction. In our example: \[ K = \frac{M v_{0}^{2}+I w^{2}}{2} \] we multiplied both sides by 2 resulting in: \[ 2K = M v_{0}^{2} + I w^{2} \] This step did not solve the problem immediately but simplified our equation by turning a fractional expression into a linear one. Once fractions are eliminated, solving the equation becomes more straightforward. The key is ensuring each term is adjusted consistently to keep the equation equal.