Problem 81 - Elementary Algebra

Question

Find the value of each of the following expressions. \(m=\frac{2 s+1}{T} . \quad\) Find \(m\) if \(s=-8\) and \(T=5\)

Step-by-Step Solution

Answer

Answer: -3

Step 1: Substitute the given values for \(s\) and \(T\)

Replace \(s\) with \(-8\) and \(T\) with \(5\) in the expression for \(m\): \(m=\frac{2(-8)+1}{5}\)

Step 2: Simplify the numerator

Calculate the numerator of the expression, which is \(2(-8)+1\): \(2(-8)=-16\) \(-16+1=-15\) Now, the expression for \(m\) is: \(m=\frac{-15}{5}\)

Step 3: Divide the numerator by the denominator

Divide the numerator \(-15\) by the denominator \(5\) to find the value of \(m\): \(m=-\frac{15}{5}=-3\) So, when \(s=-8\) and \(T=5\), the value of \(m\) is \(-3\).

Key Concepts

Substitution Method

The substitution method is a fundamental technique in algebra that involves replacing variables with given numbers or other expressions. This technique is particularly useful when trying to find the value of an expression for specific variable values.

For instance, in the exercise provided, the goal is to find the value of the expression for specific values of the variables. Here's how you can apply the substitution method effectively:

• Identify the variables in the expression and their corresponding values provided in the problem. In our example, the variables are 's' and 'T'.
• Replace each variable with its given value. This should be done carefully to avoid any errors in calculation.
• After substitution, the expression often becomes a numerical one that can be simplified through further arithmetic operations.

Understanding this technique is crucial for solving many mathematical problems and is widely used in various fields, as it bridges the gap between algebraic expressions and numerical calculations.

Simplifying Expressions

Simplifying expressions is the process of breaking down complex algebraic equations into their simplest form. It often involves combining like terms, applying arithmetic operations, and reducing fractions. Simplification makes it easier to understand, solve, and interpret mathematical problems.

Let's explore the essential steps to simplify expressions:

• Look out for opportunities to combine like terms, which often involves combining constants or coefficients with the same variable.
• Use the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to perform calculations in the right sequence.
• Reduce fractions by dividing the numerator and the denominator by their greatest common factor.

By simplifying expressions step by step, as seen in the exercise, you can make complex problems much more manageable.

Numerical Calculation

Numerical calculation involves using arithmetic to work out the values of expressions once they have been simplified down to numbers. This is the part where the abstract part of algebra becomes concrete, as you conclude with actual numerical values.

Key points in numerical calculation include:

• Follow the correct order of operations to ensure accurate results.
• Work carefully through calculations to avoid simple errors, such as misplacing a negative sign or incorrect division.
• Be mindful of the precision required for the problem; sometimes an exact number is needed, but other times an approximation or rounding may be sufficient.

Once you've mastered the art of numerical calculation as demonstrated in the solution process of the provided exercise, you'll find that solving complex equations becomes a much more straightforward task.