16.5 Addition and Subtraction Properties of Equality

Addition and Subtraction Properties of Equality state that adding or subtracting the same quantity to both sides of an equation always produces an equivalent equation, providing the foundational tool for isolating a variable by removing an added or subtracted term.

Adding the Same Quantity to Both Sides

Adding the Same Quantity to Both Sides

If the same quantity is added to both sides of a true equation, the resulting equation remains true, since adding an identical amount to two equal quantities keeps them equal to each other.

Subtracting the Same Quantity from Both Sides

Subtracting the Same Quantity from Both Sides

If the same quantity is subtracted from both sides of a true equation, the resulting equation remains true, since removing an identical amount from two equal quantities keeps them equal to each other.

What Kind of Quantity Can Be Applied

Numerical Quantity Applied to Both Sides

The quantity added or subtracted may be a simple constant number, applied identically to both sides of the equation, most commonly used to remove a constant term from the side containing the variable.

Variable Expression Applied to Both Sides

The quantity added or subtracted may also be an entire expression containing a variable, applied identically to both sides, commonly used to move a variable term from one side of the equation to the other.

Additive Inverse Use across an Equation

Subtracting a quantity from both sides is equivalent to adding its additive inverse to both sides, so the subtraction property can always be understood as a special case of the addition property applied with a negative quantity.

Confirming These Properties Preserve Equivalence

Equality Preservation after Addition

Because the same quantity is added to both sides, the difference between the two sides remains exactly what it was before the addition; if that difference was zero, meaning the original equation was true, it remains zero after adding the same amount to each side.

Equality Preservation after Subtraction

By the same reasoning, subtracting the same quantity from both sides leaves the difference between the two sides unchanged, so a true equation remains true and its full solution set is preserved after the subtraction.

Undoing These Transformations

Addition and Subtraction Transformation Reversal

Adding a quantity to both sides can always be undone by subtracting that same quantity from both sides afterward, and subtracting a quantity can always be undone by adding it back; this reversibility confirms that these transformations produce genuinely equivalent equations rather than merely similar-looking ones.