Problem 2 - Elementary Algebra

Question

For the following problems, find \(-a\) if \(a\) is: $$ -15 $$

Step-by-Step Solution

Answer

Answer: The opposite of the number -15 is 15.

Step 1: Identify the given number

The given number is \(a = -15\).

Step 2: Multiply by \(-1\)

To find \(-a\), multiply the given number \(a\) by \(-1\): $$-a = (-1) \times a$$

Step 3: Calculate the result

Now multiply \(-1\) by \(-15\) to find \(-a\): $$-a = (-1) \times (-15)$$ $$-a = 15$$ So, \(-a = 15\).

Key Concepts

Algebraic Operations

In algebra, operations such as addition, subtraction, multiplication, and division play a fundamental role in solving equations and manipulating expressions. Understanding the rules for these operations, especially when dealing with negative numbers, is crucial for success.

When working with negative numbers, it's important to remember that adding a negative number is the same as subtracting its positive counterpart, and subtracting a negative number is equivalent to adding its positive version. Simplifying algebraic expressions often requires the use of the distributive property, which involves multiplying a single term by each term inside a parenthesis. For example, in the expression \(3(x - 4)\), we would distribute the \(3\) to both \(x\) and \( -4\), resulting in \(3x - 12\).

Another key rule in algebraic operations is the concept of 'like terms', which are terms that have the same variable raised to the same power. These can be combined through addition or subtraction to simplify expressions further. For instance, \(2x + 3x = 5x\) since both terms have the variable \(x\) to the first power.

Multiplication of Integers

Multiplication of integers involves a set of rules that are easy to follow once you understand the concept of positive and negative numbers. When multiplying integers, the sign of the result depends on the signs of the numbers being multiplied together.

• If both integers are positive, the result is positive (e.g., \(3 \times 2 = 6\)).
• If one integer is positive and the other is negative, the result is negative (e.g., \( -3 \times 2 = -6\)).
• If both integers are negative, the result is positive (e.g., \( -3 \times -2 = 6\)).
• Zero multiplied by any integer is always zero (e.g., \(0 \times -5 = 0\)).

In the context of the given exercise, we're looking at \( -1 \times -15\), which according to the rules, results in a positive number because both integers are negative. This is a fundamental concept that helps explain why \( -a \), when \(a\) is negative, gives us a positive result.

Subtraction in Algebra

Subtraction in algebra can often be interpretative, especially when combined with negative numbers. The process of subtracting a negative number can seem counterintuitive at first, but it follows a logical rule that makes the arithmetic work out consistently.

In algebraic terms, subtracting a negative number is the same as adding its positive equivalent. This means we can rewrite a subtraction equation as an addition equation with the sign of the second number changed. For example, \(5 - (-3)\) is the same as \(5 + 3\), giving us \(8\). This rule helps to simplify equations and avoid errors when dealing with negative numbers.

The principle showcased in the exercise is a direct application of this rule. To find the negative of a negative number, we effectively subtract it, leading us to 'add' its positive counterpart. Thus, when we are asked to find \( -a \), and our \(a\) is \( -15\), we are performing the operation \( -(-15)\), which simplifies to \( +15\). This understanding is essential for students grappling with the concept of negative numbers in algebra.