Problem 79 - Elementary Algebra

Question

Find the value of each of the following expressions. \(z=\frac{x-u}{s} . \quad\) Find \(z\) if \(x=23, u=25,\) and \(s=1 .\)

Step-by-Step Solution

Answer

Answer: -2

Step 1: Substituting the given values

Replace \(x\), \(u\), and \(s\) in the expression with the given values: \(x=23, u=25,\) and \(s=1.\) We get: $$z=\frac{23-25}{1}$$

Step 2: Simplifying the expression

Perform the subtraction in the numerator and the division: $$z = \frac{-2}{1}= -2$$ So, the value of \(z\) is \(-2\).

Key Concepts

Substituting Variables

When solving algebraic expressions, the first step often involves substituting variables. This process replaces the letters in an equation with their corresponding numerical values. It is crucial in transforming an abstract equation into a concrete, solvable problem. For example, consider an equation like \( z = \frac{x - u}{s} \). Here, \( x \), \( u \), and \( s \) are placeholders representing unknown values.

Once these values are given, as in our exercise where \( x = 23 \), \( u = 25 \), and \( s = 1 \), we can replace the variables with their provided numbers. This results in a new expression: \[ z = \frac{23 - 25}{1} \]. It's important to replace each variable carefully to avoid errors, and exact replacement is key to finding the correct answer.

• Identify the variables in the equation.
• Replace each variable with its given value.
• Ensure each substituted value is in the correct position.

Simplifying Expressions

The process of simplifying expressions is critical for reducing complex algebraic equations to their simplest form. In our exercise, once variables are substituted, we move to simplification. This often requires performing operations such as addition, subtraction, multiplication, and division.

For the given equation \[ z = \frac{23 - 25}{1} \], the next step is to simplify the numerator by subtracting 25 from 23, yielding a result of -2. Since the denominator is 1, dividing any number by 1 yields the number itself. Therefore, we find that \[ z = \frac{-2}{1} = -2 \].

• Combine like terms and use the order of operations.
• Perform arithmetic operations such as addition and subtraction.
• Reduce fractions or terms to their lowest terms if necessary.

It is essential to approach this step by step to avoid mistakes and to make the problem easier to understand.

Numerical Operations in Algebra

A deep understanding of numerical operations in algebra is foundational for solving equations. Algebraic problems often involve numerical operations such as addition, subtraction, multiplication, and division, but also more advanced concepts like exponents and roots.

In the context of our exercise, after substituting the variable values, we performed a subtraction (\(23 - 25\)) and then a division (\(-2 ÷ 1\)) to reach the final answer \(-2\). These operations are based on standard arithmetic but are applied within an algebraic framework. This process demonstrates:

• The importance of following the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• How simple numerical operations can be used to simplify algebraic expressions, leading to the solution of an equation.
• The equivalence of certain operations, like dividing by 1, which does not change the value of a number.

Mastering these operations is fundamental to success in algebra and more advanced mathematics.