Problem 40 - Elementary and Intermediate Algebra

Question

Add. Do not use the number line except as a check. \(-25+25\)

Step-by-Step Solution

Answer

0

Step 1: Identify the Numbers

Recognize the numbers involved in the problem. We have -25 and 25.

Step 2: Understanding Addition of Opposites

When adding a number and its opposite, the result is always 0. This is because they cancel each other out.

Step 3: Apply the Rule

Since -25 and 25 are opposites, add them together: -25 + 25 = 0.

Step 4: Verify with Number Line (Optional)

As an optional check, you can use a number line. Starting at 0, move 25 units to the left to reach -25, then 25 units to the right to return to 0.

Key Concepts

Opposite numbers

Opposite numbers are pairs of numbers that are the same distance from zero but in opposite directions on a number line. In our exercise, the numbers -25 and 25 are opposites.
For example:

• The opposite of 3 is -3.
• The opposite of -7 is 7.

When we add a number and its opposite, they cancel each other out because one number is positive and the other is negative, leading to a zero sum.
This concept is crucial in understanding integer addition and is frequently used in algebra and arithmetic.

Zero sum

A zero sum occurs when the total of adding certain numbers equals zero. This happens when a number is added to its opposite.
Let's look at our example: \( -25 + 25 = 0 \).
Here, -25 and 25 are opposites and when added, their magnitudes cancel each other out, giving us a zero sum.
The principle of zero sum is useful for simplifying expressions and solving equations by eliminating terms that cancel each other.
This can also be observed in real-life scenarios, such as balancing a budget where an expense and its reimbursement cancel each other out.

Using a number line for verification

A number line is a visual tool that helps to verify integer addition. Here’s how you can use it:

• Start at zero on the number line.
• First, move left by 25 units to reach -25.
• Then, move right by 25 units to reach back to 0.

In our example, starting at 0 and moving -25 takes us to -25. Moving back 25 units brings us back to 0, confirming that -25 and 25 add up to zero.
This visualization reinforces the concept of opposite numbers and zero sum, making it easier to understand and apply these concepts in various problems.