Problem 22 - Algebra A Combined Function

Question

Solve each formula for the specified variable. \(P R=x+y+z+w\) for \(z\)

Step-by-Step Solution

Answer

\( z = PR - x - y - w \)

Step 1: Isolate Terms with z

We start with the equation \( PR = x + y + z + w \). First, we isolate the term containing \( z \) by subtracting \( x \), \( y \), and \( w \) from both sides of the equation. This gives us: \[ PR - x - y - w = z \]

Step 2: Solve for z

Since \( z \) is already isolated on one side of the equation from the previous step, we have solved the equation for \( z \). Therefore, \( z = PR - x - y - w \).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is like moving parts of an equation around to make it simpler or to solve for a specific variable. Think of it as rearranging the pieces of a puzzle to see the complete picture.

In the context of our equation, the goal is to creatively rearrange terms to focus on the variable you want to solve for, which is often not initially isolated. Good algebraic manipulation can include actions like

• Adding or subtracting the same quantity from both sides of the equation.
• Multiplying or dividing both sides by the same nonzero number.
• Using distribution and factoring to simplify expressions.

These moves maintain the balance of the equation, meaning whatever you do to one side, you have to do to the other side as well. In the given problem, we subtract others terms from both sides to keep things balanced while simplifying the equation to bring us closer to solving for the variable of interest.

Isolation of Variables

Isolation of variables is a fundamental goal when solving equations. The idea is to end up with the desired variable alone on one side of the equation, while everything else is on the opposite side. Essentially, it involves rearranging the equation until your target variable is by itself.

In the exercise at hand, our task was to solve the formula for the variable \( z \). The process begins with rearranging the equation so that \( z \) is by itself. The original equation is \( PR = x + y + z + w \). By isolating \( z \), we've effectively created a new equation, \( z = PR - x - y - w \), which is much easier to work with in further analysis, as it expresses \( z \) explicitly in terms of other known quantities.

By isolating variables, we make the solution clearer and more structured, which is vital when dealing with more complex scenarios or equations.

Formula Rearrangement

Formula rearrangement is closely tied to both algebraic manipulation and variable isolation. It involves systematically altering the structure of a formula to suit analysis or solve for a specific variable.

What you do in formula rearrangement is reshuffle the components of the equation by performing legitimate math operations, like adding, subtracting, multiplying, or dividing all sides evenly. This process ensures the equation remains balanced, which means whatever is gained on one side is equally offset on the other.

In the given equation \( PR = x + y + z + w \), formula rearrangement is used after isolating \( z \). The terms \( x \), \( y \), and \( w \) are moved to the other side of the equation through subtraction. This rearranging of terms helps in directly expressing most of the variables in terms of \( z \), which is essential for evaluations needing clear and precise variable expression. It's a strategy used frequently to not only find solutions but also to identify patterns and validate theoretical results.