Problem 11 - Algebra A Combined Function

Question

Solve each formula for the specified variable. \(f=5 g h\) for \(h\)

Step-by-Step Solution

Answer

\(h = \frac{f}{5g}\)

Step 1: Understand the Formula

The formula given is \(f = 5gh\), where \(f\) is equal to the product of \(5\), \(g\), and \(h\). We need to solve for \(h\).

Step 2: Isolate the Variable

To solve for \(h\), we need to isolate \(h\) on one side of the equation. Start by dividing both sides of the equation by \(5g\).

Step 3: Perform the Division

Divide both sides of the equation by \(5g\), resulting in \(h = \frac{f}{5g}\). This simplifies the equation, isolating \(h\).

Step 4: Final Expression

Now, the equation solved for \(h\) is \(h = \frac{f}{5g}\). This is the rearrangement of the original formula to express \(h\) in terms of \(f\) and \(g\).

Key Concepts

Solving for a Variable

In algebra, one common task is solving equations for a specific variable. This means adjusting the equation until the desired variable is on one side alone. For example, consider an equation like \( f = 5gh \). If we are asked to solve for \( h \), it means we need to change the arrangement of the equation so that \( h \) is by itself on one side. This process helps us understand how the variable relates to the others in the formula. When dealing with mathematical expressions, this technique allows us to find the value of the variable based on known quantities.

Isolation of Variables

Isolating a variable is an important step in solving equations. It involves arranging an equation so that the variable of interest is on one side of the equation by itself. In the provided exercise, we have the equation \( f = 5gh \), and we need to isolate \( h \). Start by identifying what operations are being performed on the variable. Here, \( h \) is being multiplied by \( 5g \). To isolate \( h \), we need to undo this multiplication.

Steps to Isolate a Variable

• Identify the current operations involving the variable. Here, it's multiplication by \( 5g \).
• Perform the inverse operation on both sides of the equation. For multiplication, this means division.
• Ensure the variable is left alone on one side of the equation. This gives the result \( h = \frac{f}{5g} \).

This procedure allows us to express \( h \) independently, making it straightforward to compute its value if \( f \) and \( g \) are known.

Division in Equations

Division is a fundamental mathematical operation often used to simplify equations and solve for variables. In our example, the equation \( f = 5gh \) requires division to isolate \( h \). Since the equation is in the form of multiplication, we use division to "undo" this operation. This is an application of the inverse operation principle.

Why Use Division?

• Division helps in reversing the multiplication performed on the variable.
• It simplifies the equation, making it easier to interpret and solve.

By dividing both sides of the equation by \( 5g \), the term \( h \) becomes isolated: \( h = \frac{f}{5g} \). This method ensures that the equality of the equation is maintained while rearranging it to find the desired variable. Such a division offers clarity, showing how \( h \) changes with different values of \( f \) and \( g \).