Problem 82 - Elementary Algebra

Question

Find the value of each of the following expressions. \(m=\frac{2 s+1}{T} . \quad\) Find \(m\) if \(s=-10\) and \(T=-5 .\)

Step-by-Step Solution

Answer

Answer: The value of \(m\) when \(s = -10\) and \(T = -5\) is \(\frac{19}{5}\).

Step 1: Substitute the values of s and T

Since we are given the values of \(s\) and \(T\), we will plug these values into the expression for \(m\). Doing this, we get: \(m=\frac{2(-10) + 1}{-5}\).

Step 2: Calculate the expression

Now, we will calculate the expression: $m = \frac{2(-10) + 1}{-5} \\ = \frac{-20 + 1}{-5} \\ = \frac{-19}{-5}$.

Step 3: Simplify the expression

Now we can simplify the expression: \(m = \frac{-19}{-5} = \frac{19}{5}\).

Step 4: Answer

The value of \(m\) when \(s = -10\) and \(T=-5\) is \(\frac{19}{5}\).

Key Concepts

Substitution

In algebra, substitution is the process of replacing a variable with a given number or expression in an algebraic equation or expression. By doing this, we "substitute" or "plug in" these values into the formula. For the expression \(m = \frac{2s+1}{T}\), substitution comes in handy for finding the value of \(m\) when we know the values of \(s\) and \(T\).
Let's look at how substitution works in the provided exercise:

• First, acknowledge that \(s = -10\) and \(T = -5\).
• Substitute these values into the expression for \(m\).
  This means replacing every occurrence of \(s\) in the expression with \(-10\) and every \(T\) with \(-5\).
• The expression then becomes \(m = \frac{2(-10) + 1}{-5}\).

Once substitution is complete, the resulting expression can be evaluated to solve for the desired variable. This step creates the starting point for solving the problem by focusing on direct manipulation of given values.

Simplification

Simplification is a fundamental skill in algebra, involving reducing an expression to its simplest form. After substitution, the next step is to simplify the expression. This process usually involves performing arithmetic operations and reducing fractions or terms. Let's explore the simplification process in this exercise.
First, examine the expression after substitution:

• From the substitution, we have \(m = \frac{2(-10) + 1}{-5}\).
• Calculate inside the numerator first: \(2(-10) = -20\), so the numerator becomes \(-20 + 1\).
• This simplifies to \(-19\) as the numerator.

Now that the numerator calculation is complete, we find that our expression is \(m = \frac{-19}{-5}\).
The final fraction needs one last step of simplification to turn \(m = \frac{-19}{-5}\) into \(m = \frac{19}{5}\). In this step, both the numerator and denominator are negative, so we can simplify by changing the sign. Therefore, the simplified form of the expression is positive, \(m = \frac{19}{5}\).
Simplification helps make expressions more intuitive and easier to understand, bridging the gap between raw computation and final, usable results.

Fractions

Fractions represent a part of a whole and are written as \(\frac{numerator}{denominator}\). In mathematics, understanding how to work with fractions, especially negative ones, is crucial. Let's discuss fractions in the context of simplifying and solving the exercise.
When dealing with fractions like \(m = \frac{-19}{-5}\):

• The numerator is \(-19\) and the denominator is \(-5\).
• Notice that both the numerator and denominator are negative, which means the fraction itself is positive.
• This leads to \(m = \frac{19}{5}\) after simplification.

Here, knowing that a fraction with both a negative numerator and negative denominator equals a positive fraction is vital.
This understanding plays a significant role in solving and simplifying problems with fractions, allowing you to find correct and logical solutions.