The equality properties for addition, subtraction, multiplication, and division are fundamental principles of algebra that allow us to manipulate equations in a consistent and valid manner. These properties state that if two quantities are equal, then performing the same operation on both quantities will maintain that equality. Here’s a brief overview of each property:
- Addition Property of Equality: If \( a = b \), then \( a + c = b + c \) for any number \( c \).
- Subtraction Property of Equality: If \( a = b \), then \( a - c = b - c \) for any number \( c \).
- Multiplication Property of Equality: If \( a = b \), then \( a \times c = b \times c \) for any number \( c \).
- Division Property of Equality: If \( a = b \) and \( c \neq 0 \), then \( a / c = b / c \).
These properties work for equalities (equations) because they are based on the definition of equality itself: If two quantities are equal, any transformation that we apply to one side of the equation must also be applied to the other side to preserve that equality. This is fundamental to the field of mathematics and ensures that the operations we perform on equations do not change the truth of the original statement.
When solving for \( x \) in an equation, we apply these properties to isolate \( x \). For example, consider the equation:
\[ x + 3 = 7 \]
To solve for \( x \), you can use the Subtraction Property of Equality:
- Subtract 3 from both sides: \[ x + 3 - 3 = 7 - 3 \] \[ x = 4 \]
The operation (subtraction) is applied to both sides, maintaining the equality throughout the process.
Thus, we have confidence in these properties because they are built upon the foundational definition of equality, which states that if two things are equal, they remain equal even after performing the same mathematical operations on both sides.