Problem 64 - Elementary Algebra

Question

For the following problems, solve each literal equation for the designated letter. \(P=2(1+w)\) for \(w\).

Step-by-Step Solution

Answer

Question: Solve the equation \(P = 2(1 + w)\) for the variable \(w\). Answer: \(w = \frac{P - 2}{2}\)

Step 1: Expand the equation

First, we will expand the equation by multiplying the outer term with the terms inside the parentheses: \((2*1)+(2*w)\).

Step 2: Simplify the equation

Now that the equation has been expanded, we simplify it by adding the numbers and keeping the term with \(w\): \(P=2+2w\)

Step 3: Isolate the variable \(w\)

In order to isolate the variable \(w\), we need to move the constant term on the other side of the equation. Subtract \(2\) from both sides: \(P-2 = 2w\)

Step 4: Solve for \(w\)

Finally, to find the value of \(w\), we need to get rid of the coefficient in front of the variable \(w\). Divide both sides of the equation by \(2\): \(w=\frac{P-2}{2}\) So, the literal equation solved for the designated letter, \(w\), is: \(w=\frac{P-2}{2}\).

Key Concepts

Solving Equations

Solving equations is all about finding the value of an unknown variable that makes the equation true. In literal equations, unlike numerical ones, we deal with multiple variables. This can initially feel a bit tricky. However, the process isn't all that different from solving more straightforward numerical equations.
The key steps involve following a logical sequence of operations:

• Identify: Determine which variable to solve for, based on the given problem.
• Transform: Apply operations such as addition, subtraction, multiplication, and division to simplify the equation.
• Solve: Continue manipulating the equation until the target variable is isolated on one side, with its solution clear on the other.

In the original exercise, our objective was to find the value of \(w\). By carefully applying algebraic steps, we ensured each operation led us closer to isolating \(w\) and reaching the solution.

Algebraic Manipulation

Algebraic manipulation is the backbone of solving equations. It involves rearranging and simplifying expressions to reveal unknown values. Involves the use of basic mathematical operations in a strategic manner.

• Distributive Property: In our example, we started by expanding \(P = 2(1 + w)\). Using the distributive property, we multiplied \(2\) by both \(1\) and \(w\) to simplify the equation to \(P = 2 + 2w\).
• Simplification: This step included combining like terms, simplifying any arithmetic, and making the equation easier to work with.
• Reverse Operations: These operations are crucial to manipulate the equation in the reverse direction of its current operation to isolate the variable.

Understanding and mastering algebraic manipulation helps in solving not only literal equations but many complex algebra problems effectively.

Variable Isolation

Variable isolation is the process of arranging an equation so that the desired variable stands alone on one side of the equation. In the exercise provided, the ultimate goal was to express \(w\) in terms of other variables and constants.

• Subtracting: We moved the terms not involving the target variable to the opposite side. Specifically, \(2\) was subtracted from \(P\), which adjusted our equation to \(P - 2 = 2w\).
• Dividing: The final step involved dividing both sides by \(2\) to solve for \(w\). This operation canceled out the coefficient of \(w\), leaving it isolated as \(w = \frac{P - 2}{2}\).

Each step in isolating the variable should be clear, methodically breaking down the equation until the variable is isolated and the solution is derived. Mastering variable isolation is key to tackling literal equations and finding solutions effectively.