# Substitution (Special Cases)

We already studied this method in the section*Substitution*. Here, we will practice more and learn a couple special cases.

**Strategy:**In a problem with two variables, say,

*x*and

*y*, you must check the case in which

*x* = *y*. (This often gives a double case.)

*x*and

*y*are positive.

Column A | Column B |

Average of x and y |
Average of x^{3} and y^{3} |

Let *x* = *y* = 1. Then Column A becomes . And Column B becomes . In this case, the columns are equal. But if *x* = *y* = 2, then Column A becomes and Column B becomes . In this case, the columns are unequal. This is a double case and therefore the answer is (D).

*x*and

*y*are integers greater than or equal to 1.

Column A | Column B |

2^{x}^{ + y} |
2 + 2^{x}^{y} |

If *x* â‰ *y*, then Column A is larger than Column B. (Plug in a few numbers until you are convinced.) But if *x* = *y* = 1, then the columns are equal: 2^{x}^{ + y} = 2^{1 + 1} = 2^{2} = 4 and 2* ^{x}* + 2

*= 2*

^{y}^{1}+ 2

^{1}= 4. Hence, there is not enough information to decide, and the answer is (D).

**Strategy:** When you are given x < 0, you must plug in negative whole numbers, negative fractions, and â€“1. (Choose the numbers â€“1, â€“2, and â€“1/2, in that order.)

*k*< 0

Column A | Column B |

0 |

If *k* is â€“1 or â€“2, then Column A is larger since it is a product of squares. But if *k* = â€“1/2, then the two columns are equal:

Hence, there is not enough information to decide and the answer is (D).

**Strategy: **Sometimes you have to plug in the first three numbers (but never more than three) from a class of numbers.

*x*is both an integer and greater than 1. Let [

*x*] stand for the smallest positive integer factor of

*x*not equal to 1.

Column A | Column B |

[x] |
[x^{3}] |

Choose *x* = 2, 3, and 4. If *x* = 2, then [*x*] = 2 and [*x*^{3}] = [8] = 2. So for this choice of *x*, the two columns are equal. If *x* = 3, then [*x*] = 3 and [*x*^{3}] = [27] = 3, again the columns are equal.

Finally, If *x* = 4, then [*x*] = 2 and [*x*^{3}] = [64] = 2, still again the columns are equal. Hence, the answer is (C).

Note, there is no need to check *x* = 5. The writers of the test do not change the results after the third number.