Problem 110 - Algebra A Combined Function

Question

Subtract. $$ 5-(-10) $$

Step-by-Step Solution

Answer

The result is 15.

Step 1: Understand the Subtraction Problem

The problem given is to subtract \(-10\) from \(5\). In mathematical terms, this is written as \(5 - (-10)\). Subtracting a negative number is equivalent to adding its positive counterpart.

Step 2: Convert to Addition

Based on the rule that subtracting a negative is the same as adding a positive, transform the expression \(5 - (-10)\) to \(5 + 10\).

Step 3: Perform the Addition

Now, simply add the two numbers: \(5 + 10\). Adding these together gives you \(15\).

Key Concepts

Addition with Negative Numbers

When dealing with negative numbers, addition can initially seem tricky. But understanding a few key concepts makes it manageable. Suppose we have a simple problem where we add a negative number like \(-3\) to a positive number like \(7\). Instead of thinking of it as addition, it might help to think of it as subtraction. Imagine you are starting at 7 and then taking 3 steps backwards on the number line:

• The problem \(7 + (-3)\) becomes \(7 - 3\).
• This is because adding a negative number is the same as subtracting its positive form.
• The result is \(4\). Starting at 7 and taking 3 backwards steps lands you at 4.

Recognizing that adding a negative number is equivalent to subtraction can simplify problems significantly.
Moreover, when subtracting, like in \(5 - (-10)\), swap the minus for a plus:

• The negative sign changes direction, converting the problem to \(5 + 10\).
• This concept applies universally, as subtracting a negative turns into an addition of the equivalent positive number, simplifying calculations.

Basic Arithmetic Operations

Arithmetic operations such as addition, subtraction, multiplication, and division form the backbone of mathematics. Subtraction acts as the opposite operation to addition. Performing these operations with negative numbers requires understanding different rules.
Consider subtraction: \(a - b\) is mathematically the same as \(a + (-b)\). This interchange means that subtraction can always be transformed into addition of negative numbers:

• For instance, \(5 - 3\) is equivalent to \(5 + (-3)\).
• An equation involving two negatives, such as \(5 - (-10)\), is transformed into an addition operation, \(5 + 10\).
• This transformation property helps simplify many subtraction problems into more manageable addition problems.

To effectively work with these operations, remember:

• Addition results in the combination of quantities, while subtraction involves taking away.
• Understanding these conversions ensures easier manipulation and solves complex problems by simplifying them into basic additions.

Number Line in Algebra

A number line is a visual tool used in mathematics to represent numbers in a linear form, stretching infinitely in both directions. It is particularly helpful for understanding operations with negative numbers:

• The center of a number line is 0. Numbers increase to the right (positive) and decrease to the left (negative).
• When you add a positive number, you move right; when you add a negative or subtract a positive, you move left.

Visualize this with \(5 - (-10)\):

• You start at 5 and, since you effectively add after converting the minus sign, move 10 units to the right, landing on 15.
• Using the number line, conversion from subtraction to addition becomes clear.
• This method offers not only clarity but reinforces understanding of arithmetic sign rules.

Number lines make solving algebraic problems more intuitive. Looking at changes this way assists greatly in building a foundational understanding of how numbers behave.