Problem 44 - Elementary and Intermediate Algebra

Question

Solve each formula for the specified variable. $$ G=U-T S+P V \text { for } S $$

Step-by-Step Solution

Answer

The variable \( S \) is given by \( S = \frac{U + PV - G}{T} \).

Step 1: Understanding the equation

We need to solve the equation \( G = U - TS + PV \) for the variable \( S \). This means we need to isolate \( S \) on one side of the equation.

Step 2: Rearranging the equation

First, subtract \( U + PV \) from both sides of the equation to get: \[ G - U - PV = -TS \]

Step 3: Isolating the variable

Now, we need to solve for \( S \). To isolate \( S \), divide both sides of the equation by \(-T\): \[ S = \frac{G - U - PV}{-T} \]

Step 4: Simplifying the equation

Since dividing by a negative is equivalent to multiplying by a negative, we can rewrite the expression: \[ S = -\frac{G - U - PV}{T} \] or alternatively \[ S = \frac{U + PV - G}{T} \]

Key Concepts

Isolating Variables

Isolating a variable in an algebraic equation is a crucial concept in solving problems where you need a specific unknown part of the formula. In our exercise, the main goal is to isolate the variable \(S\). Isolating a variable means restructuring the equation so that \(S\) stands alone on one side. This involves undoing any arithmetic operations around \(S\). For example, if \(S\) is multiplied by another number, you'll often need to perform the opposite operation, such as division, to clear it. In doing so, you must apply the same operations across the entire equation to maintain balance.

• Think of the equation as a balance scale, where each side must remain equal.
• To isolate \(S\), decide which terms directly affect \(S\) and work on removing them using their inverse operations.
• Remember that operations like addition have subtraction as inverse, and multiplication has division.

For our equation \(G = U - TS + PV\), to isolate \(S\), first eliminate any terms which don't involve \(S\) directly, such as \(U\) and \(PV\), by using subtraction and division.

Rearranging Equations

Rearranging equations involves systematically moving terms from one side of an equation to the other. This helps to position the variable you're solving for in a more convenient place within the equation. Let’s break it down with our example, \( G = U - TS + PV \). Here, \( S \) is mixed with several operations and other variables.

• Begin by identifying terms you need to move. In our case, start with subtracting non-\(S\) terms by performing subtraction on both sides. We subtract \(U + PV\) from both sides to clear these terms.
• This results in the equation \( G - U - PV = -TS \), which confines \(S\) to one side.
• Next, reorganize the equation so \(S\) is isolated as much as possible by dividing or altering the position of accompanying factors.

The key is to maintain the equation’s integrity while maneuvering components, ensuring each step is legal and logical, keeping the equation balanced.

Solving for a Variable

Once you've performed the rearranging and isolations, the next logical step is solving for your targeted variable. Solving for a variable is typically the finale where the variable is expressed as a standalone term on one side of the equation.With the equation now as \( G - U - PV = -TS \), the last job is to solve \( S \) by clearing it of coefficients or factors on its side. Here, the coefficient is \(-T\).

• Divide every term of the equation by \(-T\) to isolate \( S \).
• Performing this division, you directly solve for \( S \): \( S = \frac{G - U - PV}{-T} \).
• You can rewrite this as \( S = -\frac{G - U - PV}{T} \) to further simplify, making it appear cleaner and sometimes more intuitive.

Solving for a variable is seamlessly handled by maintaining order and clarity during rearrangement, ensuring each step moves you closer to isolating the variable fully. Understanding these concepts gives you a tactical advantage in manipulating even the most complex equations with confidence.