In the realm of mathematics, particularly in algebra and geometry, the Addition and Subtraction Postulates are pivotal principles. These postulates pertain to manipulating expressions, equations, and inequalities in order to isolate and solve for variables. Here are the definitions for the Addition & Subtraction Postulates:
1. Addition Postulate
If a = b, and c = d, then a + c = b + d
If equal quantities are added to equal quantities, the sums are equal.
If congruent segments are added to congruent segments, the sums are congruent.
If congruent angles are added to congruent angles, the sums are congruent.
2. Subtraction Postulates
If a = b, and c = d, the a – c = b – d
If equal quantities are subtracted from equal quantities, the differences are equal.
If congruent segments are subtracted from congruent segments, the differences are congruent.
If congruent angles are subtracted from congruent angles, the differences are congruent.
Algebraic Context:
The Addition and Subtraction Postulates serve as key tools to solve equations and inequalities in algebra.
Example:
Consider the equation \(2x + 7 = 15\).
To find the value of \(x\), utilize the Subtraction Postulate to subtract 7 from both sides of the equation:
\[ 2x = 8 \]
Next, divide by 2 to isolate \(x\):
\[ x = 4 \]
Angles Context
In geometry, especially when dealing with angles, the Addition and Subtraction Postulates enable us to determine the measure of unknown angles given the measures of others.
Example:
If \(m\angle ABC + m\angle CBD = 180^\circ\) and \(m\angle ABC = 75^\circ\), determine the measure of \(m\angle CBD\).
Applying the Subtraction Postulate:
\[ m\angle CBD = 180^\circ – m\angle ABC \]
\[ m\angle CBD = 180^\circ – 75^\circ \]
\[ m\angle CBD = 105^\circ \]
Segments Context
In the scenario of segment measures in geometry, the Addition and Subtraction Postulates are instrumental in finding unknown segment lengths given the lengths of other segments.
Example:
Given that segment \(AB + BC = 15\) cm and \(AB = 6\) cm, find the length of segment \(BC\).
Employing the Subtraction Postulate:
\[ BC = AB + BC – AB \]
\[ BC = 15 \, \text{cm} – 6 \, \text{cm} \]
\[ BC = 9 \, \text{cm} \]