Problem 62 - Elementary Algebra

Question

Find the value of \(z\) if \(z=\frac{x-u}{s}\) and \(x=55, u=49,\) and \(s=3\).

Step-by-Step Solution

Answer

Answer: 2

Step 1: Write down the given formula and values

We are given that \(z = \frac{x-u}{s}\). We are also given the following values: 1. \(x = 55\) 2. \(u = 49\) 3. \(s = 3\)

Step 2: Plug in the given values

Now, we will plug the given values into the formula for \(z\): \(z = \frac{x-u}{s} = \frac{55-49}{3}\)

Step 3: Simplify the expression

Let's first subtract inside the numerator: \(z = \frac{6}{3}\) Now, divide the numerator by the denominator: \(z = 2\) So, the value of \(z\) is 2.

Key Concepts

Equations

In algebra, equations are fundamental tools for expressing relationships between different quantities. An equation is essentially a mathematical statement that asserts the equality of two expressions. It involves an equal sign, such as the formula given in the exercise: \[z = \frac{x-u}{s}\]This equation captures the relationship between four variables (

• \(z\), the dependent variable,
• \(x\), \(u\), and \(s\), the independent variables.

By providing specific values for these independent variables, we can find the corresponding value for the dependent variable, resulting in a clear solution to the equation. Understanding and manipulating equations are keys to solving countless algebraic problems.

Problem Solving

Problem solving in algebra often involves a series of strategic steps to find the solution of an equation. In our exercise, the problem was to find the value of \(z\) using a specified formula. This process was broken down into clear steps:

• First, identify the formula used and note any given values. This ensures you understand what is required and what you already have.
• Next, substitute the given values into the equation. Ensure that each value correctly replaces its corresponding variable.
• Finally, simplify the expression by performing the necessary arithmetic operations. This can include addition, subtraction, multiplication, and division.

This methodological approach is crucial in algebra, as it encourages clarity and ensures that the solution is logically derived from known quantities. Problem-solving techniques are not only important for getting the correct answer but also for developing analytical skills that are useful in numerous real-world scenarios.

Variables

Variables represent unknown or changeable values in an equation. In our exercise, \(x\), \(u\), and \(s\) were variables with given values, whereas \(z\) was the unknown we needed to solve for. Variables are symbols that replace numbers in equations and can take various values depending on the context.There are different types of variables:

• Independent variables: These are the variables you can change, such as \(x\), \(u\), and \(s\) in the equation. They influence the dependent variable but are not affected by it.
• Dependent variables: This is the output of the equation, which depends on the values of independent variables. In our case, \(z\) is the dependent variable as its value is derived from the values of \(x\), \(u\), and \(s\).

Understanding the role of variables enables us to manipulate equations effectively and is essential for exploring and solving diverse mathematical problems. With practice, handling variables becomes intuitive, allowing for more complex problem solving in algebra and beyond.