Problem 80 - Elementary Algebra

Question

Find the value of each of the following expressions. \(z=\frac{x-u}{s} . \quad\) Find \(z\) if \(x=410, u=430,\) and \(s=2.5 .\)

Step-by-Step Solution

Answer

Question: Given the expression \(z = \frac{x-u}{s}\), find the value of \(z\) when \(x = 410\), \(u = 430\), and \(s = 2.5\). Answer: -8

Step 1: Substitute the given values

Begin by substituting the given values for \(x, u,\) and \(s\) into the expression: $$z = \frac{410 - 430}{2.5}$$

Step 2: Perform the subtraction

Next, subtract \(430\) from \(410\) in the numerator: $$z = \frac{-20}{2.5}$$

Step 3: Perform the division

Finally, divide \(-20\) by \(2.5\): $$z = -8$$ Therefore, the value of \(z\) is \(-8\).

Key Concepts

Substitution Method

The substitution method is an essential technique in algebra that involves replacing variables with their corresponding numerical values. This process is crucial when solving algebraic expressions or equations, as it allows you to transform an abstract formulation into a calculable numerical expression.

For example, consider the algebraic expression for computing the value of a variable such as: \(z = \frac{x - u}{s}\). When the values for the variables \(x, u,\) and \(s\) are provided, one can determine \(z\) by substituting the given numbers directly into the expression. This method not only simplifies the process but also helps in visualizing how each variable affects the overall value of the expression. Effective substitution requires careful attention to replacing each variable with its exact numerical counterpart and performing the operations in the correct order.

Numerical Expression Evaluation

Evaluating a numerical expression involves carrying out the operations within an expression step by step, according to the order of operations. This order dictates that calculations are to be performed from left to right, following: parentheses, exponents, multiplication and division (from left to right), and finally, addition and subtraction (from left to right).

In our exercise example, once the variables are substituted, the expression \(z = \frac{410 - 430}{2.5}\) can be evaluated. Starting with the subtraction in the numerator yields \(z = \frac{-20}{2.5}\), and proceeding with the division as the final step results in \(z = -8\). It's crucial to carry out these operations correctly to avoid errors in the final result. The main goal is to simplify the expression to the point where no more operations are needed, leaving you with a numerical value that is the solution to the given problem.

Algebraic Problem-Solving

Algebraic problem-solving involves breaking down and solving mathematical problems that require finding unknown values through the manipulation of algebraic expressions. This process typically includes several stages, such as understanding the problem, devising a plan with an appropriate method (like substitution), carrying out the plan, and then reviewing the solution for accuracy.

Looking again at the given example, the problem is already understood to find the value of \(z\). The plan involves using the substitution method and then employing the correct numerical expression evaluation techniques. After performing the substitution and calculations, one should double-check the arithmetic and ensure that each operation has been executed in proper sequence. Effective algebraic problem-solving implies a methodical approach, sound reasoning, and frequently, a bit of trial and error to ensure the most accurate outcome.