Problem 40 - Elementary and Intermediate Algebra

Question

Solve each formula for the specified variable. $$ P=2 l+2 w \text { for } l $$

Step-by-Step Solution

Answer

\( l = \frac{P - 2w}{2} \)

Step 1: Identify the formula and the variable to solve for

The given formula is \( P = 2l + 2w \), and we need to solve for the variable \( l \). This means we want to express \( l \) in terms of the other variables in the equation.

Step 2: Isolate terms involving the desired variable

To solve for \( l \), we first need to isolate the term that contains \( l \). In the equation \( P = 2l + 2w \), we'll separate \( 2l \) by subtracting \( 2w \) from both sides: \[ P - 2w = 2l \]

Step 3: Solve for the variable

Now that we have isolated the term \( 2l \), we solve for \( l \) by dividing both sides of the equation by 2: \[ l = \frac{P - 2w}{2} \]

Step 4: Verify the solution

Verify by substituting \( l = \frac{P - 2w}{2} \) back into the original equation. Substitute and simplify to ensure both sides are equal, confirming the solution is correct.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is the process of modifying algebraic expressions or equations to achieve a desired form. It often involves applying mathematical operations like addition, subtraction, multiplication, division, and factoring. The primary goal is usually to simplify the expression or to solve for a variable of interest. In our example with the equation \( P = 2l + 2w \), the aim is to resolve the expression such that we can express \( l \) by itself.Key operations for algebraic manipulation include:

• Combining like terms: Terms that have identical variables raise to the same power can be combined.
• Using the distributive property: This involves expanding expressions like \( a(b + c) \) to \( ab + ac \).
• Rearranging the equation: Moving terms from one side of the equation to the other by performing inverse operations helps to isolate a specific variable.

By effectively using these operations, we can transform complex formulas into more manageable forms.

Formulas in Algebra

Formulas in algebra are equations that express relationships between different variables. They are pivotal in solving problems where one quantity depends on others. A formula can be thought of as a mathematical recipe describing how different quantities interact.For instance, the formula \( P = 2l + 2w \), which represents the perimeter \( P \) of a rectangle, relates the perimeter (P) to its length (l) and width (w). The formula tells us that if we know any two of these values, we can find the third.Common formulas seen in algebra:

• Area and perimeter of geometric shapes: Different shapes have specific formulas to compute their area and perimeter.
• Linear equations: These describe a straight line on a coordinate plane and are generally expressed as \( y = mx + b \).
• Quadratic equations: These are polynomial equations of the second degree, often found in the form \( ax^2 + bx + c = 0 \).

Understanding formulas is instrumental as they provide a framework for solving real-world problems mathematically.

Isolation of Variables

Isolation of variables refers to the technique of rearranging an equation so that a specific variable stands alone on one side. This is crucial when you want to solve for a particular variable in terms of others.In our example formula \( P = 2l + 2w \), to solve for \( l \), we isolate it by:

• Subtracting \( 2w \) from both sides to get \( P - 2w = 2l \).
• Dividing each side by 2, resulting in \( l = \frac{P - 2w}{2} \).

Steps in isolating variables effectively:

• Identify the term with the desired variable and aim to get it on one side of the equation.
• Use inverse operations (addition or subtraction, multiplication or division) to isolate the variable.
• Simplify the equation at every step to make calculations easier.

Isolation is foundational as it allows us to express relationships between variables in a more useful and understandable form.