Use identities to simplify an expression. For example, the expression (X^2 + Y^2) / (X + Y) becomes simpler if you use the identity X^2 + Y^2 = (X - Y)(X + Y) for substitution. (X^2 + Y^2) / (X + Y) = (X - Y)(X + Y) / (X + Y) = X - Y. Similarly, the expression (X^3 + Y^3) / (X + Y) becomes simpler when you use the identity X^3 + Y^3 = (X + Y)(X^2 + XY + Y^2) for substitution. (X^3 + Y^3) / (X + Y) = (X + Y)(X^2 + XY + Y^2) / (X + Y) = X^2 + XY + Y^2.
Solve simultaneous equations with substitution. Consider the word problem: "A woman is five times as old as her son. In 10 years, she will be three times as old as her son. How old is the son now?" Clearly W = 5S and W + 10 = 3(S + 10) = 3S + 30. If we substitute the first equation into the second equation we get an equation we can solve: (5S) + 10 = 3S + 30. This equation has only one unknown and we can solve it to get S = 10. The son is 10 years old.
Make up a substitution if an identity or simultaneous equation is not available. For example, solving the equation X^4 - 8X^2 + 16 can seem daunting if you only know how to solve quadratic equations. Letting Y = X^2 provides a substitution that allows you to turn X^4 - 8X^2 + 16 into the quadratic Y^2 - 8Y + 16. This factors into Y^2 - 8Y + 16 = (Y - 4)^2. This has the solution Y = 4. Going back to the original equation, there are two solutions X= +2 and X = -2.