The symmetric property of equality states that if two variables a and b exist, and a = b, then b = a. The symmetric property of equality is one of the equivalence properties of equality.
The symmetric property of equality allows individuals to manipulate an equation by flipping the statements on each side of the equals sign. For example, in algebra, this means that the equations 11 = 2x + 5 and 2x + 5 = 11 are equivalent.
One useful application of the symmetric property of equality is that reorganizing equations makes it easier to solve systems of equations. For example, if one had two linear equations, 11 = 2x + 5 and -2x + 3y = 6, the symmetric property of equality allows the equations to be ordered so that they can be added together. Thanks to the symmetric property of equality, 11 = 2x + 5 can be transformed to say 2x + 5 = 11. One can add the two equations together to solve for y: (-2x + 3y = 6) + (2x + 5 = 11). These two equations sum to 3y + 5 = 17.
This simplifies to 3y = 12, or y = 4. Then by substituting in for y, -2x + 3(4) = 6; this yields -2x + 12 = 6, or -2x = -6, or x = 3. So the solution to the linear system is x = 3, y = 4.