Lesson 68: Algebraic Addition

New Concept

This course builds a bridge from arithmetic to algebra. We will learn to see numbers and operations in new, more powerful ways to solve problems.

What’s next

We'll begin with our first algebraic idea: reframing subtraction. Next, you'll explore worked examples showing how to simplify expressions using algebraic addition.

Property

A positive number and a negative number whose absolute values are equal are opposites.

Examples

• The opposite of 8 is $-8$.
• The opposite of $-25$ is $25$.
• The opposite of $\frac{3}{4}$ is $-\frac{3}{4}$.

Explanation

Imagine a number line is a street and zero is your house. Opposites are like two friends who live the same distance from your house but in opposite directions! So, 7 is seven steps to the right, and its opposite, -7, is seven steps to the left. They're mirror images across zero, perfectly balanced every single time.

Property

To use algebraic addition, treat subtraction as adding the opposite. The problem $a - b$ becomes an addition problem: $a + (-b)$.

Examples

• $15 - 5$ can be solved by thinking $15 + (-5) = 10$.
• $-9 - 4$ is simplified by thinking $(-9) + (-4) = -13$.
• $8 - (-3)$ becomes an addition problem: $8 + [-(-3)] = 8 + 3 = 11$.

Explanation

Think of this as a math superpower! Instead of taking numbers away, you can turn any subtraction problem into an addition party just by adding the opposite number. This trick makes simplifying expressions with lots of negative signs way easier to handle. It turns a confusing subtraction mess into a straightforward addition problem every single time.

Property

Taking the opposite of an opposite brings you back to the original number.

Examples

• The expression $-(-11)$ simplifies directly to $11$.
• To solve $-5 - (-9)$, we think addition: $(-5) + [-(-9)] = (-5) + 9 = 4$.
• Simplify $[-(-20)] + (-12)$ becomes $20 + (-12) = 8$.

Explanation

Think of the minus sign as an 'opposite' command that makes you turn around 180 degrees. If you get two 'opposite' commands in a row, you'll spin around twice and end up facing the same direction you started! So, the opposite of negative five, or $-(-5)$, just cancels out and brings you right back to 5.