Unit I: Introduction to Algebra

The Problem: The Mystery Box

Imagine you have a scale. On one side, there's a mystery box of unknown weight, plus 3 one-pound weights. On the other side, there are 7 one-pound weights. The scale is perfectly balanced. How much does the mystery box weigh?

You can probably solve this in your head, but what you're doing is algebra! You're solving for an unknown. To do this systematically, mathematicians developed a language using symbols. They replaced the "mystery box" with a letter, like x.

This turns our problem into an equation: a statement that two things are equal.

The Golden Rule of Algebra

The goal of algebra is to get the mystery box (x) by itself on one side of the scale. The "Aha!" moment is realizing the one, unbreakable rule: Whatever you do to one side of the equation, you MUST do to the other side to keep it balanced.

Use the simulator below to solve different equations. Start with x + 3 = 7. Try removing 3 blocks from the left side. What happens? Now remove 3 from the right side as well to find the value of x.

Isolating the Variable

By removing the same number of '1' blocks from both sides, you successfully isolated 'x' and kept the scale balanced. You were left with the mystery box on one side and the remaining blocks on the other, revealing the value of x. You've solved your first equation!

This "balancing" method is the foundation of all algebra. To solve for a variable, you use inverse operations (like subtraction to undo addition) on both sides of the equation until the variable is by itself. This reveals its value.

Key Takeaways

• Variable: A letter (like 'x') used to represent an unknown number.
• Equation: A statement that two expressions are equal, like a balanced scale.
• The Golden Rule: To keep an equation balanced, whatever operation you perform on one side, you must also perform on the other.