Problem 30 - Elementary Algebra

Question

Use numerical evaluation to evaluate the equations for the following problems. $$ \begin{aligned} &z=\frac{x-u}{s} . \quad \text { Find } z \text { if } x=42, u=30, \text { and }\\\ &s=12 \end{aligned} $$

Step-by-Step Solution

Answer

Question: Given the equation $$z = \frac{x - u}{s}$$, find the value of z when x = 42, u = 30, and s = 12. Answer: The value of z is 1.

Step 1: Substitute the given values

Replace the variables x, u, and s with their given numerical values: 42, 30, and 12, respectively, in the equation $$z = \frac{x - u}{s}$$. Thus, the equation becomes: $$z = \frac{42 - 30}{12}$$,

Step 2: Perform subtraction in the numerator

Subtract the two numbers in the numerator: $$42 - 30 = 12$$ Now the equation becomes: $$z = \frac{12}{12}$$

Step 3: Divide the numerator by the denominator

Divide 12 by 12 to find the value of z: $$z = \frac{12}{12} = 1$$ So, the value of z is 1.

Key Concepts

Variables Substitution

Variables substitution is a crucial process in solving equations. In this exercise, we had an equation with variables \(x\), \(u\), and \(s\). Substituting means replacing these variables with their given values to simplify and solve the equation. For our problem, we replaced \(x\), \(u\), and \(s\) with 42, 30, and 12, respectively. This turned our algebraic equation \(z = \frac{x-u}{s}\) into a numerical equation \(z = \frac{42-30}{12}\).
This step is essential because it allows us to work with concrete numbers instead of abstract symbols, making the equation easier to solve.

• Identify the variables in the equation.
• Replace each variable with its given value.
• Simplify the equation if possible.

Subtraction

In mathematics, subtraction involves taking one number away from another. During this exercise, the subtraction operation was performed in the numerator of our fraction \(\frac{42-30}{12}\).
We subtracted 30 from 42, resulting in 12, which simplified our expression to \(\frac{12}{12}\).
Understanding subtraction is key in problems like these, as it allows us to reduce complex parts of the equation into easier, more manageable pieces.

• Ensure subtraction is performed at the right place, often within the numerator or when simplifying expressions.
• Check your work to avoid mistakes - even small ones can drastically change solutions.

Division

Division is another fundamental operation in mathematics. In our exercise, after performing subtraction, we needed to divide \(12\) by \(12\).
Division tells us how many times one number fits into another. In this case, \(12\) divides \(12\) exactly once, giving us the result \(z = 1\).
When dividing numbers, especially in the context of algebra, it’s important to simplify completely to find the final solution.

• A division of equal numbers (like 12 divided by 12) yields 1.
• Always perform division when simplifying expressions step-by-step.
• Check that the denominator is not zero to avoid undefined results.

Algebraic Equations

Algebraic equations like \(z = \frac{x-u}{s}\) are a key part of mathematics. They are statements of equality containing variables and operations. Solving these equations provides a value for the unknown variable. In our case, we were finding the value of \(z\).
Here’s a simple approach to handling algebraic equations:

• Start by substituting given values for known variables.
• Simplify by performing arithmetic operations (like subtraction and division) as needed.
• Ensure each step logically follows the last, checking for accuracy throughout.

Algebraic equations can represent real-world phenomena, where understanding how different elements interact through mathematical expressions is valuable. This exercise demonstrates how numerical evaluation helps us find precise solutions.