Question

Find the sum. $$-7+7$$

Short answer

The sum of -7 and 7 is 0.

Step-by-step solution

Identifying the Numbers

Identify the numbers in the exercise, -7 and 7.

Adding the Numbers

Add the two numbers together. In this case, since -7 and 7 are opposites, they will cancel each other out to become 0.

Key concepts

Adding Opposites

When you come across a math problem that involves adding opposites, you're dealing with two numbers that are the same distance from zero on the number line, but in opposite directions. One will have a negative sign (-), and the other will have a positive sign (+).

For example, in the exercise where we find the sum of \( -7 + 7 \), -7 and 7 are opposites. Imagine you're standing on a number line: if you take 7 steps to the left (representing -7) and then 7 steps to the right (representing +7), you end up back at your starting point, which is zero. This is why when you add -7 and 7, the result is 0.

Understanding this concept is important not just for simple arithmetic, but also for algebra and higher-level mathematics, where you'll frequently encounter variables that can represent positive and negative numbers.

Integers Sum

The sum of integers refers to the result you get when you add two or more integers together. Integers are whole numbers that can be either positive, negative, or zero. When adding integers, it's helpful to think of the number line again. Positive integers move you to the right, while negative integers move you to the left.

If you add a positive integer to another positive integer, the sum will be positive (\(3 + 5 = 8\)). The same goes for two negative integers—the sum will be negative (\( -3 + (-5) = -8 \)). However, when you add a positive and a negative integer together, they may cancel each other out if they are exact opposites, like in our originally given exercise (\( -7 + 7 = 0 \)).

Using a Number Line for Positive and Negative Sums

Visualizing these operations on a number line can help make the concept clearer and highlight the symmetry between the positive and negative sides of zero.

Zero Pairs

In mathematics, the concept of zero pairs is incredibly useful, especially when working with integers. A zero pair consists of one positive and one negative number that, when added together, equal zero. Think of them as pairs that cancel each other out.

In our exercise, \( -7 \) and \(+7\) form a zero pair. You can imagine zero pairs like objects that create a balance. If you have a negative apple (\( -1 \) apple) and a positive apple (\(+1\) apple), together they don't contribute to the total count of apples; they neutralize each other, resulting in zero apples.

Why Zero Pairs Matter?

Recognizing zero pairs is particularly helpful when simplifying algebraic expressions or solving equations. They help you quickly identify terms that will eliminate each other, making the problems less complex and easier to solve.