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Chapter 3: Problem 53

Solve the equation \(z=\frac{x-m}{s}\) for the variable \(m\) using the given values of \(z, s,\) and \(x\) $$z=1.28, s=3.8, \text { and } x=10.164$$

Short Answer

Expert verified
The value of \( m \) is 5.3.

Step by step solution

01

Understand the initial equation

We start with the equation given in the problem: \( z = \frac{x-m}{s} \). This equation defines \( z \) as a standardized score.
02

Substitute the known values

We substitute the known values into the equation: \( z = 1.28 \), \( x = 10.164 \), and \( s = 3.8 \). This results in the equation \( 1.28 = \frac{10.164 - m}{3.8} \).
03

Solve for m

Begin by eliminating the fraction. Multiply both sides of the equation by \( s \) to clear the fraction: \( 1.28 \times 3.8 = 10.164 - m \). Simplify the left side to find \( 4.864 = 10.164 - m \).
04

Isolate m

To solve for \( m \), subtract \( 10.164 \) from both sides of the equation: \( 4.864 - 10.164 = -m \). This simplifies to \( -5.3 = -m \).
05

Solve for m

Since \( -5.3 = -m \), we can multiply both sides by \( -1 \) to find \( m = 5.3 \). Thus, we have the value of \( m \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations are fundamental in algebra and represent expre鈥媠sions where the variable is raised to the first power only. In a linear equation, the relationship between variables is directly proportional, meaning if one variable increases, the other changes at a constant rate. Given the equation \( z = \frac{x-m}{s} \), we are dealing with a simple linear expression where \( z \) is related to \( x \), \( m \), and \( s \) through operations of subtraction and division.
  • Understanding the structure of linear equations is crucial because it tells us that variables can be isolated using basic algebraic operations.
  • It's important to identify which variable you are solving for, which in this case is \( m \).
Linear equations can be manipulated through addition, subtraction, multiplication, and division, allowing us to isolate the desired variable.
Standardized Score
A standardized score, often denoted as \( z \), is a way of scaling individual data points to a common standard. This allows for comparison across different data sets. The formula \( z = \frac{x-m}{s} \) reflects how far and in what direction a data point \( x \) deviates from the mean \( m \), measured in terms of the standard deviation \( s \).
  • Think of the standardized score as a universal translator for data points, which makes comparisons simpler and clearer.
  • In the given problem, \( z = 1.28 \) suggests that the data point \( x = 10.164 \) is 1.28 standard deviations away from the mean \( m \).
Using standardized scores helps in understanding distributions and identifying where a particular value stands in the dataset.
Solving for Variables
Solving for variables is a common task in algebra, requiring the isolation of one variable so that its value can be determined. In the equation \( 1.28 = \frac{10.164 - m}{3.8} \), solving for \( m \) involves a series of methodical steps:
  • First, eliminate fractions by multiplying both sides by the denominator, \( s \), to simplify the equation.
  • Next, rearrange the equation by performing algebraic operations like addition or subtraction to isolate \( m \).
  • Finally, ensure the variable is completely isolated, which might involve multiplying or dividing both sides by a constant.
This structured approach allows you to systematically find the value of any variable by keeping the equation balanced. It's a fundamental skill in algebra, crucial for solving equations effectively and confidently.

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Most popular questions from this chapter

Write the direct variation equation, determine the constant of variation, and then calculate the indicated value. Round to three decimal places as necessary. \(y\) varies directly with \(x\) and \(y=21\) when \(x=3\). Find \(y\) when \(x=42\).

The average length of a prison sentence, measured in years, in 2012 can be determined by solving the equation $$1.6874=m-2.58(4.7)$$ Solve for \(m,\) round to the nearest year, and interpret the result.

Write the direct variation equation, determine the constant of variation, and then calculate the indicated value. Round to three decimal places as necessary. \(a\) varies directly with \(b\) and \(a=12\) when \(b=3\). Find \(a\) when \(b=12\).

Solve the absolute value inequality, write the answer in interval notation, and graph the solution on the real number line. $$|x| < 4$$

Solve the equation \(z=\frac{x-m}{s}\) for the variable \(s\) using the given values of \(z, x,\) and \(m\). $$z=2, x=10, \text { and } m=8$$
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