Problem 41 - Intermediate Algebra

Question

Solve each formula for the specified variable. See Example 5. \(V=\) Iwh for \(h\)

Step-by-Step Solution

Answer

The solution for \( h \) is \( h = \frac{V}{Iw} \).

Step 1: Identify the Equation and Variables

The given formula to solve is \( V = Iwh \). We are tasked with solving for the variable \( h \). "Solve for \( h \)" means we need to express \( h \) in terms of the other variables in the equation: \( V \), \( I \), and \( w \).

Step 2: Isolate the Variable

To isolate \( h \), divide both sides of the equation by the product \( Iw \) (the coefficient of \( h \)). This gives us:\[ h = \frac{V}{Iw} \] This equation now gives \( h \) in terms of \( V \), \( I \), and \( w \).

Key Concepts

Isolating Variables

Isolating a variable in an equation is a crucial step in solving for that variable. It involves manipulating the equation so that the variable you are solving for is by itself on one side of the equation. In the exercise provided, the goal is to solve for the variable \( h \) in the equation \( V = Iwh \). The first step in isolating \( h \) involves understanding that it is currently multiplied by both \( I \) and \( w \). Therefore, to isolate \( h \), you need to perform the opposite operation of multiplication, which is division.
This means dividing both sides of the equation by \( Iw \). By doing so, you achieve:

• Moving the terms \( Iw \) from the right side of the equation to underneath \( V \) on the left side
• Getting the variable \( h \) alone on one side, effectively isolating it

Isolating variables enables clear solutions and is an essential skill in algebra, giving clarity to what each variable represents in terms of others.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying equations using various algebraic techniques. It enables one to rearrange terms and solve for specific variables. In our exercise, we applied algebraic manipulation to solve the equation \( V = Iwh \) for \( h \). This manipulation required understanding that the operations done to one side of an equation must also be done to the other side to maintain balance. For example, when we divided \( V \) by \( Iw \), we also divided \( Iwh \) by \( Iw \) to keep the equation balanced.
This particular approach used in manipulation is inverse operations:

• When terms are added, you subtract to move terms to the other side
• When terms are multiplied, you divide to move terms to the other side

Such techniques are central in algebra to simplify and solve equations efficiently. Mastering these skills allows us to handle complex equations in a systematic and logical manner.

Variables in Equations

Variables in equations are symbols that represent unknown values or numbers. They are the elements that change and interact with each other according to the rules specified in an equation. In our original exercise, the variables were \( V \), \( I \), \( w \), and \( h \). Their relationship was defined by the equation \( V = Iwh \), which is akin to saying that the volume \( V \) is a result of multiplying the current \( I \), width \( w \), and height \( h \).
Understanding variables requires knowing that:

• Each variable can represent many different numbers or values
• Adjusting one variable can affect others, hence impacting the entire equation or system
• Each variable holds specific meaning within the context of the problem

Comprehending how variables interact in equations allows you to manipulate them correctly to find solutions. It provides a fundamental building block for solving mathematical problems not only in algebra but across various fields of study.