Problem 68 - Algebra and Trigonometry with Analytic Geometry

Question

The formula occurs in the indicated application. Solve for the specified variable. \(R=\frac{V}{I}\) for \(I\)

Step-by-Step Solution

Answer

The solution is \(I = \frac{V}{R}\).

Step 1: Identify the Given Formula

We are given the formula \(R = \frac{V}{I}\) where \(R\) is resistance, \(V\) is voltage, and \(I\) is current. Our goal is to solve for the variable \(I\).

Step 2: Rearrange the Formula

To isolate \(I\), we need to manipulate the equation. Start by multiplying both sides of the equation by \(I\): \(I \cdot R = V\).

Step 3: Solve for the Variable

Now, solve for \(I\) by dividing both sides of the equation by \(R\): \(I = \frac{V}{R}\).

Key Concepts

Solving Equations

Solving equations is a fundamental aspect of mathematics and science. It involves finding the value of an unknown variable that makes an equation true. In many practical applications, such as in physics and engineering, solving equations is crucial.
Understanding how to manipulate equations allows you to derive specific information from general formulas. In this exercise, for example, you need to express one variable in terms of others, which often involves multiple steps of arithmetic processes.
To solve for a particular variable:

• Identify the equation and the variable to be solved for.
• Perform operations such as addition, subtraction, multiplication, or division to both sides of the equation to isolate the variable.
• Check your solution by substituting the found variable back into the original equation to ensure it holds true.

Practicing these techniques will improve your ability to intuitively manipulate and solve equations in different contexts.

Rearranging Formulas

Rearranging formulas is a skill that enables you to convert an equation from one form to another. This is particularly useful when you need to determine the value of a specific variable from a given relationship. In the exercise presented, we start with the formula for Ohm's Law:
Given: \[ R = \frac{V}{I} \]We aim to solve for \(I\), the current. Rearranging formulas often involves the following steps:

• Identify which operations are affecting the variable you want to isolate and perform inverse operations to move other terms away.
• Be systematic, with each step bringing you closer to the desired form.
• Always work algebraically sound by performing the same operation on both sides of the equation to maintain equality.

In this case, multiplying both sides by \(I\) gives \( I \cdot R = V \), and dividing by \(R\) gives the isolated form: \( I = \frac{V}{R} \). By mastering rearranging techniques, you can derive different forms of relations and apply them in diverse situations.

Electrical Circuits

Electrical circuits are systems of connected components used to conduct and control the flow of electricity. Understanding how these circuits work involves knowing the relationships between voltage \(V\), current \(I\), and resistance \(R\), which are elegantly described by Ohm's Law.
Ohm’s Law states that \( V = I \times R \). It is a fundamental principle used in circuit analysis to calculate the current, voltage, or resistance when the other two are known, similar to how you solve the formula in the given exercise.
In practical terms:

• Voltage (\(V\)) is the potential difference that motivates charge flow.
• Current (\(I\)) is the rate at which charge flows through a surface.
• Resistance (\(R\)) is the opposition to current flow within the circuit.

Understanding electrical circuits and using Ohm’s Law is crucial for designing and troubleshooting circuit systems, as it allows you to predict and calculate the electrical behavior of the system effectively.