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Chapter 4: Problem 14

Solve each equation. $$\text { Solve } P=L+\frac{s}{f} i \text { for } i$$

Short Answer

Expert verified
The solution to the equation is \(i = \frac{(P - L) * f}{s}\)

Step by step solution

01

Subtract \(L\) from both sides of the equation

Subtract \(L\) from both sides to isolate the term with \(i\) on one side of the equation: \(P - L = \frac{s}{f} i\).
02

Isolate \(i\)

Multiply both sides of the equation by \(f\) and divide by \(s\) to completely isolate \(i\):\(i = \frac{(P - L) * f}{s}\)Now, \(i\) is expressed in terms of the other variables.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Manipulation
Algebraic manipulation is about rearranging equations or expressions to make them easier to solve or understand. In solving equations, this often involves using operations such as addition, subtraction, multiplication, and division to transform the equation's structure.
It helps in simplifying equations or making a particular variable more apparent, especially when dealing with complex expressions.
To manipulate the given equation, we start by performing operations that maintain the equality of both sides. These operations allow us to change the equation in a way that brings us closer to our goal, which in this case, is solving for the variable \(i\).
  • Firstly, subtract \(L\) from both sides to simplify the expression: \(P - L = \frac{s}{f}i\).
  • This step eliminated \(L\) from the right side of the equation and helped isolate the term that contains \(i\).
Isolation of Variables
Isolation of variables is a key strategy in solving equations. It means rearranging the equation so that the variable you are solving for stands alone on one side. This is a crucial step because it allows us to clearly see the solution and understand the relationship between the variables involved.
In our example, we aim to solve for the variable \(i\). After we have simplified the initial equation, we focus on getting \(i\) by itself on one side. To achieve this, you should:
  • Identify terms that need to be moved or changed to isolate your target variable.
  • Use inverse operations鈥� the opposite of what is being done to the variable鈥攖o gradually isolate it.
After subtracting \(L\), we have \(P - L = \frac{s}{f}i\). Multiplying both sides by \(f\) enables us to cancel out \(f\) from the denominator, aiding in this isolation process.
Equation Solving Techniques
Equation solving techniques for algebra often require a mix of strategies to find the solution in an efficient manner. These techniques include employing operations and transformations to alter the equation's form while ensuring its equality remains unchanged.
The main goal is to find an expression for the unknown variable (in this case \(i\)), in terms of the known quantities. Let鈥檚 break down these techniques:
  • Subtraction Method: When terms need to be removed from one side, subtraction is often used. This was shown in our first step to balance the equation.
  • Multiplication/Division Method: To isolate variables within fractions, multiplication and division are utilized to cancel unwanted terms, just like multiplying by \(f\) in our equation.
These algebraic techniques are fundamental to solving any kind of equation, making this skill critical for understanding and solving mathematical problems efficiently.

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