Problem 49 - Elementary and Intermediate Algebra

Question

Solve each formula for the specified variable. $$ l=a+(n-1) d \text { for } d $$

Step-by-Step Solution

Answer

\( d = \frac{l-a}{n-1} \)

Step 1: Understand the equation

We are given the equation \( l = a + (n-1)d \) and need to solve for \( d \). This means we need to rearrange the formula to isolate \( d \) on one side of the equation.

Step 2: Isolate the term containing d

Subtract \( a \) from both sides of the equation to start isolating the term containing \( d \): \( l - a = (n-1)d \).

Step 3: Solve for d

To solve for \( d \), divide both sides of the equation by \( n-1 \). This gives: \[ d = \frac{l-a}{n-1} \]. Now, \( d \) is isolated.

Key Concepts

Isolate Variable

When solving equations, one of the primary skills to master is isolating the variable of interest. This is all about making sure the variable you're solving for is all by itself on one side of the equation. To do this, you will need to carefully perform inverse operations.

For instance, in the equation \( l = a + (n-1)d \), if the goal is to isolate \( d \), you need to eliminate other elements gradually until \( d \) stands alone. Start by removing terms that are added or subtracted from the side with \( d \). If it's being multiplied, dividing will cancel out the multiplication.

**Quick Tips:**

• Look for terms you can subtract or add.
• Don't forget inverse operations like division for multiplication.
• Double-check by substituting back to see if the results make sense.

Keeping these tips in mind allows you to systematically solve for any variable in linear equations.

Linear Equations

Linear equations are equations of the first degree, meaning they have variables raised only to the power of one. These equations form straight lines when graphed and are foundational to understanding algebra.

In processing a linear equation, think about its main form, represented as \( ax + b = c \). Each part of the equation can tell you where to start. Often, these equations will simply need a rearranging of terms to find the desired variable.

**Why They Matter:**

• Easy to solve: Require basic arithmetic operations like addition, subtraction, multiplication, and division.
• Forms the building blocks for more complex algebraic concepts.
• Applicable in countless real-world situations such as budgeting and calculating distances.

Understanding the nature and behavior of linear equations is crucial as you delve deeper into algebra and other mathematical areas.

Rearranging Formulas

Rearranging formulas is a critical skill in algebra. It's essentially the art of manipulating the equation to get your desired variable isolated, much like transforming ['l=a+(n-1)d'] to solve for \( d \).

During rearrangement, the goal is to make adjustments that keep the equation balanced. Here’s how you can think of it:

**Steps for Effective Rearranging:**

• Identify the target variable you need to solve for.
• Use inverse operations to move other terms across the equation.
• Be systematic: It’s often helpful to perform a series of smaller steps rather than tackling everything at once.
• Check as you go to ensure no mistakes are made.

Rearranging formulas is about understanding the role of each term and knowing which steps will most effectively isolate the variable of interest. The practice pays off in flexibility and precision.