Problem 75 - Elementary Algebra

Question

Solve for the indicated variable. $$ \text { Solve for } b: \quad P=a+b+c $$

Step-by-Step Solution

Answer

\(b = P - a - c\)

Step 1: Understand the Equation

We are given the equation \( P = a + b + c \) and need to solve for the variable \( b \). This means we need to express \( b \) in terms of the other variables \( P, a, \) and \( c \).

Step 2: Isolate the Variable b

To solve for \( b \), we need to isolate it on one side of the equation. We start with the equation \( P = a + b + c \). Subtract \( a \) and \( c \) from both sides of the equation: \( P - a - c = b \).

Step 3: Simplify the Equation

The equation \( b = P - a - c \) is already simplified. This is the expression for \( b \) in terms of the other variables.

Key Concepts

Variable Isolation

Variable isolation is a crucial step in solving algebraic equations. It involves manipulating the equation so that the variable you are solving for stands alone on one side of the equation. Here, the goal is to solve for the variable \( b \) in the equation \( P = a + b + c \). To achieve this, you need to move all other terms to the opposite side of the equation. By subtracting \( a \) and \( c \) from both sides, you effectively "isolate" the variable \( b \). This operation transforms the original equation into \( b = P - a - c \). The process can be broken down as follows:

• Start with the equation \( P = a + b + c \).
• To isolate \( b \), subtract \( a \) from both sides, resulting in \( P - a = b + c \).
• Next, subtract \( c \) from both sides, giving you \( P - a - c = b \).

By systematically moving terms across the equation with opposite operations, you successfully isolate \( b \). This makes solving or analyzing the equation much simpler.

Equation Simplification

Equation simplification is used to make equations easier to interpret and solve. Once a variable has been isolated, the equation often becomes simpler, as in the example \( b = P - a - c \). This is where we ensure that the isolated variable is presented in its simplest form. Simplification involves combining like terms, reducing expressions, and ensuring there are no further calculations needed on the isolated side.In the given exercise, after isolating \( b \), you can quickly see there are no further terms to combine or reduce. Hence, the simplified result \( b = P - a - c \) clearly states the relationship between \( b \) and the other variables \( P \), \( a \), and \( c \). By verifying that you have simplified each part as much as possible, you confirm that the equation is at its most understandable and practical form for use in problem-solving.

Solving for Variables

After isolating and simplifying, the final step is solving for the desired variable within the context of a given problem. In algebra, 'solving' typically means finding the value of the variable. However, in a scenario where the solution is in terms of other variables, such as \( b = P - a - c \), the solution signifies expressing the variable in relation to others.Key points when solving for variables:

• Ensure the variable is isolated on one side, as in variable isolation.
• Simplify the expression, combining like terms and reducing complexity where possible.
• Always double-check your work to confirm that the variable expression logically matches the original equation.

Solving for a variable in terms of others allows you to understand the relationship and dependency between variables. Moreover, this approach is great for scenarios where you know values for most variables, but need to determine a specific one based on the equation.