91Ó°ÊÓ

In mathematics, a system of equations refers to a set of equations with multiple variables that are solved together. The objective is to find a common solution for all the variables. In our problem involving resistors, we have a system of equations due to the relationship between voltage (V), current (I), and resistance (R). The problem gives us two scenarios, each with its own equation:

• Scenario 1: 3.00 A passes through both resistors, creating two equations: \( V_1 = 3R_1 \) and \( V_2 = 3R_2 \) with a combined voltage of 10.5 V.
• Scenario 2: 2.00 A and 4.00 A pass through the resistors, respectively, giving us equations: \( V_1 = 2R_1 \) and \( V_2 = 4R_2 \) with a total voltage of 13.0 V.

By using these equations, we set up a system to solve for \( R_1 \) and \( R_2 \). This system can be mathematically represented and solved using methods such as Gaussian elimination, substitution, or graphing.

The voltage-current relationship in an electrical circuit is fundamental in understanding how circuits work. Ohm's Law states that the voltage across a resistor equals the product of the current flowing through it and its resistance. This can be written as:\[ V = I imes R \]This formula is crucial when solving electric circuits since it directly relates the three major parameters: voltage (V), current (I), and resistance (R). In our exercise, this relationship helps us to set up equations that model each scenario of current passing through the resistors. Knowing either two of the three parameters allows us to calculate the third, and this is where our systems of equations find their utility.

Resistors are components in a circuit that impede the flow of electric current. Calculating resistor values involves using the voltage-current relationship. In our exercise, the task is to find the resistances, \( R_1 \) and \( R_2 \), given various currents and voltages:

• First, express the voltages through the resistors using Ohm's Law: \( V_1 = I_1R_1 \) and \( V_2 = I_2R_2 \).
• Then, use the given conditions about voltage sums to form equations.

The equations derived from these conditions allow us to solve for \( R_1 \) and \( R_2 \). Understanding the role of resistors in an electrical system, as well as their calculation, provides insight into how circuits can be designed to control current and voltage.

Gaussian elimination is a systematic way of solving systems of linear equations. Here’s how it is applied in our exercise:- **Step 1: Understand the problem**: Identify what is given and what needs to be solved. We have resistors with given currents and known voltage sums.- **Step 2: Set up equations**: Formulate the mathematical model using the given conditions: - For voltages with current 3.00 A, we have equations: \( V_1 = 3R_1 \) and \( V_2 = 3R_2 \). - For conditions with current 2.00 A and 4.00 A, the equations are: \( V_1 = 2R_1 \), \( V_2 = 4R_2 \).- **Step 3: Simplification**: Rewrite the equations to simplify calculations. Divide equation 1 to reduce to \( R_1 + R_2 = 3.5 \).- **Step 4: Eliminate variables**: Use Gaussian elimination to transform the system and solve for one variable at a time.- **Step 5: Solve**: Substitute back to find the other unknown, ensuring all conditions are satisfied.These steps ensure a structured approach to solving the exercise, reflecting how mathematical theory applies to practical electrical problems.