Problem 131 - Beginning and Intermediate Algebra

Question

Write each of the following as an expression and evaluate. The sum of \(-2\) and the quotient of \(-15\) and 3

Step-by-Step Solution

Answer

The expression evaluates to -7.

Step 1: Define Expression Components

Identify the components needed for the expression. We need the sum of -2 and the quotient of -15 and 3.

Step 2: Calculate the Quotient

Calculate the quotient of -15 and 3. This is done by dividing -15 by 3.\[ \frac{-15}{3} = -5 \]

Step 3: Formulate the Expression

Formulate the mathematical expression based on the problem. The expression is the sum of -2 and the quotient calculated in the previous step.\[ -2 + (-5) \]

Step 4: Evaluate the Expression

Evaluate the expression formulated in the previous step. This involves adding -2 to -5.\[ -2 + (-5) = -7 \]

Key Concepts

Evaluating Expressions

Evaluating expressions means finding what value an expression represents. Think of it as solving a puzzle where you put all the pieces together to find the answer. In the provided exercise, we needed to evaluate the sum of two parts:

• The first part was the number \(-2\).
• The second part was the result of a division (also called a quotient).

To successfully evaluate the expression, it's important to know the correct order of operations. This ensures you perform mathematical operations in the right sequence to get the correct answer.
Here's a simple way to remember the order: PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
This keeps everything in the right order as you solve the expression, leading you to the correct answer.

Quotients

Quotients are the result you get when you divide one number by another. It’s that simple! In mathematics, division is symbolized by \(\div\) or by a fraction line \(\frac{\underline{\phantom{xx}}}{\underline{\phantom{xx}}}\). In our example, the quotient was part of the expression we had to evaluate.
When we divided \(-15\) by \(3\), we were seeking out the quotient of these numbers. So our task was to calculate \(\frac{-15}{3}\). Here's what happens:

• Negative divided by positive gives a negative result.
• \(15\) divided by \(3\) is \(5\).
• Thus, \(\frac{-15}{3} = -5\).

This quotient plays a crucial role in evaluating the full expression, as it's part of the sum we needed to evaluate. It's important to carefully compute quotients, especially when negative numbers are involved.

Negative Numbers

Negative numbers can sometimes seem tricky, but once you know the rules, they become much easier to handle. Negative numbers are simply numbers less than zero, represented with a minus sign \(-\). They’re the opposite of positive numbers.
In our expression, we worked with negative numbers in two key ways:

• -2 is part of the sum we needed to evaluate.
• The calculated quotient, -5, was also negative.

When adding or subtracting negative numbers, think of it like gaining and losing items. Adding a negative number is like going further back on a number line, equivalent to subtraction. For example, adding \(-5\) to \(-2\) changes the problem into a straightforward subtraction: \(-7\).
Understanding how negative numbers interact with each other through basic operations is fundamental to algebra, especially when evaluating expressions.