Problem 45 - Elementary and Intermediate Algebra

Question

Solve each formula for the specified variable. $$ z=\frac{x-\mu}{\sigma} \text { for } x $$

Step-by-Step Solution

Answer

The solution for \( x \) is \( x = z \cdot \sigma + \mu \).

Step 1: Understand the Formula

The given formula is \( z = \frac{x - \mu}{\sigma} \). This is a rearrangement of the standard score formula, often used in statistics to denote how many standard deviations an element \(x\) is from the mean \(\mu\). You are solving for \(x\).

Step 2: Eliminate the Fraction

To isolate \(x\), first eliminate the fraction by multiplying both sides of the equation by \(\sigma\). This results in \( z \cdot \sigma = x - \mu \).

Step 3: Solve for x

Now that we have \( z \cdot \sigma = x - \mu \), add \(\mu\) to both sides to solve for \(x\). This gives \( x = z \cdot \sigma + \mu \).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of an unknown variable. Think of it as a puzzle where you need to figure out what number the variable represents. In algebra, equations are like statements that declare two things are equal, with one or more unknowns to solve for. Usually, you will see an equals sign (=) between two expressions. When solving an equation, the main goal is to isolate the variable you're solving for. While dealing with any equation, always ensure the steps taken maintain the balance of the equation. If you perform an operation on one side, you must perform it on the other side too. This keeps the equation fair and correct. To solve an equation:

• Identify the variable to solve for.
• Remove or simplify any fractions or complicated expressions around the variable.
• Use subtraction, addition, multiplication, or division to isolate the variable on one side of the equation.
• Ensure all operations respect the equality by doing the same to both sides.

Remember, the operations you use will depend on the equation you're working with, and sometimes, you might need more than one step to isolate and solve for the variable.

Standard Score Formula

The standard score formula, often referred to as the z-score, is a statistical tool used to determine how far away a data point is from the mean of a data set, measured in standard deviations. Understanding this formula is crucial for data analysis and helps in comparing results from different data sets.The formula is expressed as:\[ z = \frac{x - \mu}{\sigma} \]Where:

• z is the z-score or standard score, indicating how many standard deviations the element is from the mean.
• x represents the data point in question.
• \mu (mu) is the average or mean of the data set.
• \sigma (sigma) is the standard deviation, a measure of the data's dispersion.

This formula helps in standardizing any data point within a set, making it easier to understand its relative position. For example, a z-score of 0 means the data point is exactly at the mean, while a positive or negative z-score indicates a deviation above or below the mean, respectively. This is particularly useful in fields such as psychology, finance, and any area that relies on statistical analysis.

Rearranging Equations

Rearranging equations is the process of altering the structure of an equation to solve for a specific variable. It's an essential skill because it allows you to manipulate equations to make them more usable or appropriate for solving a problem.In the context of the standard score formula given, the goal was to solve for the variable x from the equation:\[ z = \frac{x - \mu}{\sigma} \]To rearrange this equation:

• First, eliminate the fraction by multiplying both sides by \sigma, giving us z \cdot \sigma = x - \mu.
• Next, isolate x by adding \mu to both sides, resulting in x = z \cdot \sigma + \mu.

These steps demonstrate the logical process of "undoing" the operations around x to isolate it. Understanding how to rearrange equations not only helps in algebra but is also useful in real-life situations, such as financial calculations, scientific problems, and any field that involves variable analysis.