lesson
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Imagine a balanced scale holding a mystery gift box and 5 one-dollar coins on the left pan, perfectly balancing 12 one-dollar coins on the right pan.
If you take 5 coins off the left pan, how many coins must you remove from the right pan to keep the scale perfectly level?
📊A clean 2D vector graphic of a balance scale. Left pan has a mystery box labeled 'x' and five gold coins with '+5'. Right pan has 12 gold coins labeled '12'. Red curved arrows show 5 coins lifting off both sides simultaneously. Below, a balanced scale shows just 'x' on the left and '7' gold coins on the right. Modern flat design, soft shadows, #1e2945 text, #f8f9fa background.
Can we turn this simple balancing trick into an algebraic rule that works every single time?
The Balance Principle
An equation is a statement that two mathematical expressions have the exact same value, connected by an equal sign (=).
To isolate a variable means getting that single letter completely by itself on one side of the equal sign with a coefficient of 1.
To isolate a variable, we use inverse operations, which are opposite operations that undo each other—like addition undoing subtraction.
How do we undo a number that is being added to our variable?
The Subtraction Property of Equality
The Subtraction Property of Equality states that if a=b, then a−c=b−c. Subtracting the same amount from both sides preserves the balance.
Let's solve x+14=39. Since 14 is added to x, we subtract 14 from both sides: x+14−14=39−14, giving x=25.
📊A step-by-step algebraic card showing vertical alignment: Top row: 'x + 14 = 39'. Middle row in red/coral accent: ' - 14 - 14'. A horizontal dividing line. Bottom row in vibrant blue: 'x = 25'. To the side, a green check box: 'Check: 25 + 14 = 39 ✓'. Clean, high-contrast typography.