Problem 49 - Introductory Chemistry Atoms First

Question

Given \(P+Q=z\), solve for \(P\).

Step-by-Step Solution

Answer

Given the equation \(P + Q = z\), we solve for \(P\) by subtracting \(Q\) from both sides, which gives us \(P = z - Q\).

Step 1: Write down the given equation

First, we write down the given equation as it is: \(P + Q = z\).

Step 2: Subtract Q from both sides

In order to isolate \(P\), we will subtract \(Q\) from both sides of the equation. This cancels out \(Q\) on the left side of the equation: \[ P + Q - Q = z - Q \]

Step 3: Simplify the equation

Now, let's simplify the equation. Since \(Q - Q\) equals zero, we are left with: \[ P = z - Q \]

Step 4: Write down the final answer

We have successfully isolated \(P\) on one side of the equation, and our final answer is: \[ P = z - Q \]

Key Concepts

Equation Solving

When confronted with an equation like \( P + Q = z \), the goal is to solve for a specific variable, in this case, \( P \). The process of equation solving involves finding the value of a variable that makes the equation true. In practice, this means performing operations that simplify the equation until the variable of interest stands alone on one side.
To solve equations, you commonly apply inverse operations. These operations, like addition and subtraction or multiplication and division, undo each other. For this equation, solving involves strategic steps to shift all terms not containing \( P \) to the opposite side.

• Look closely at operations within the equation. Here, addition is initially used.
• Use subtraction, the inverse operation, to cancel terms as needed.

Understanding equation solving builds a foundation for more complex problem-solving tasks and helps in developing logic skills.

Isolating Variables

Isolating a variable is one of the crucial skills in basic algebra. The primary aim here is to have the variable of interest by itself on one side of the equation. When we isolate \( P \) in the equation \( P + Q = z \), our strategy involves moving \( Q \) to the other side.
The best way to do this is by performing the opposite operation that’s currently affecting the variable. For instance:

• Here, \( Q \) is added to \( P \). Therefore, subtract \( Q \) from both sides to neutralize its effect on \( P \).
• Ensure you perform operations equally on both sides to maintain the equation’s balance.

After isolating the variable, you'll find that the expression across the equation represents the variable's value. This is how we find that \( P = z - Q \). Isolating variables is a foundational skill that simplifies understanding algebraic relationships.

Algebraic Manipulation

Algebraic manipulation is the art of rewriting expressions to achieve a desired form. This could mean rearranging terms, factoring, expanding, or simplifying expressions like \( P + Q = z \). It involves making strategic decisions on what operations to apply to rearrange terms effectively.

Key techniques in algebraic manipulation include:

• Adding or subtracting terms to both sides, allowing for term removal or relocation.
• Combining like terms to streamline the equation.

With our equation, the algebraic manipulation involved subtraction of \( Q \) from both sides, leading to a simplified form of \( P = z - Q \). This step is pivotal for transforming equations into an interpretable format. Enhance your algebra skills by practicing manipulation in varied contexts.