Problem 4 - Fundamentals of Mathematics

Question

For problems \(4-9,\) find the value of each expression. $$ 6 a-3 b, \text { if } a=-2, \text { and } b=-1. $$

Step-by-Step Solution

Answer

The value of the expression is \(-9\).

Step 1: Substitute the values of variables

Given the expression \(6a - 3b\) and the values \(a = -2\) and \(b = -1\), substitute these values into the expression. So, replace \(a\) with \(-2\) and \(b\) with \(-1\). The expression becomes \(6(-2) - 3(-1)\).

Step 2: Multiply the constants and variables

Calculate \(6(-2)\) and \(-3(-1)\) separately. First, perform the multiplication for the first term: \(6 \times -2 = -12\). Then, calculate the second term: \(-3 \times -1 = 3\).

Step 3: Simplify the expression

Combine the calculated terms: \(-12 + 3\).

Step 4: Perform the final calculation

Add \(-12\) and \(3\) to get the result: \(-12 + 3 = -9\).

Key Concepts

Substitution Method

The substitution method is a way to solve mathematical expressions by replacing variables with specific values. This method is particularly helpful when you have clearly defined values for variables and need to evaluate expressions quickly. In our exercise, we are given the expression \(6a - 3b\) with specific values for \(a\) and \(b\).

• Start by identifying the variables in the expression: in this case, \(a\) and \(b\).
• Next, replace these variables with their given values: \(a = -2\) and \(b = -1\).

So, the expression \(6a - 3b\) becomes \(6(-2) - 3(-1)\) after substitution.
By substituting, you transform the expression into one without variables, making it much easier to simplify.

Order of Operations

When solving equations or expressions, it's crucial to follow the order of operations to ensure correct results. The order of operations can be remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

• First, tackle any operations inside Parentheses.
• Next, solve any Exponents if they are present.
• Then, handle Multiplication and Division as they appear from left to right.
• Lastly, perform any Addition and Subtraction from left to right.

In our scenario, multiplication comes first since there are no parentheses or exponents in \(6(-2) - 3(-1)\). Thus, you multiply \(6\) by \(-2\), and \(-3\) by \(-1\), resulting in \(-12\) and \(3\) respectively. The final step involves adding these results together: \(-12 + 3\).
Following these steps ensures that each part of the expression is handled correctly and yields the proper solution.

Integer Arithmetic

Understanding integer arithmetic is essential when dealing with expressions involving negative and positive numbers. Integers are whole numbers that can be positive, zero, or negative, and knowing how to operate with them simplifies math problems. When you multiply or divide two integers:

• The product or quotient of two positive integers is positive.
• The product or quotient of two negative integers is also positive.
• The product or quotient of one positive and one negative integer is negative.

In the problem \(6(-2) - 3(-1)\), you have:

• \(6 \times -2\), resulting in \(-12\) since a positive and a negative number produce a negative result.
• \(-3 \times -1\), resulting in \(3\) as two negative numbers produce a positive result.

Finally, adding \(-12\) and \(3\) gives you \(-9\). By mastering these basic rules of integer arithmetic, you can easily tackle similar math problems.