# Properties of Equality

The following are the properties of equality for real numbers . Some textbooks list just a few of them, others list them all. These are the logical rules which allow you to balance, manipulate, and solve equations.

 PROPERTIES OF EQUALITY Reflexive Property For all real numbers $x$ , $x=x$ . A number equals itself. These three properties define an equivalence relation Symmetric Property For all real numbers $x\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}y$ , if $x=y$ , then $y=x$ . Order of equality does not matter. Transitive Property For all real numbers $x,y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}z$ , if $x=y$ and $y=z$ , then $x=z$ . Two numbers equal to the same number are equal to each other. Addition Property For all real numbers $x,y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}z$ , if $x=y$ , then $x+z=y+z$ . These properties allow you to balance and solve equations involving real numbers Subtraction Property For all real numbers $x,y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}z$ , if $x=y$ , then $x-z=y-z$ . Multiplication Property For all real numbers $x,y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}z$ , if $x=y$ , then $xz=yz$ . Division Property For all real numbers $x,y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}z$ , if $x=y$ , and $z\ne 0$ , then $\frac{x}{z}=\frac{y}{z}$ . Substitution Property For all real numbers $x\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}y$ , if $x=y$ , then $y$ can be substituted for $x$ in any expression. Distributive Property For all real numbers $x,y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}z$ , $x\left(y+z\right)=xy+xz$ For more, see the section on the distributive property