By Aristotle

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- Representation theory of Hopf Categories
- Smoothness, Regularity and Complete Intersection
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1 + n + m + z2 86. n(m + z 2 ) 13 35 89. 0 The expressions in Problems 90–93 can be put in the form ax2 + x. For each expression, identify a and x. 90. bz 2 + z 92. t2 +t 7 (a) 2 adults and 3 kids (b) No adults and 4 kids (c) No kids and 5 adults (d) A adults and C kids 1 + 2r + 3s Apago PDF Enhancer 84. 101. A family of two adults and three children has an en- 1+r+s rs 1+n+m+z 85. (n + m)z 2 87. (a) 2 adults and 3 kids (b) No adults and 4 kids (c) No kids and 5 adults (d) A adults and C kids 91.

4 to obtain the second equation. (b) We subtract y 3 from both sides of the first equation to get the second. Apago PDF Enhancer Warning: Dividing Both Sides by an Expression That Might Be Zero Not every operation that we can perform on both sides of an equation leads to an equivalent equation. If you divide both sides by an expression that could be equal to zero, then you might lose some solutions. Example 4 What operation transforms the first equation into the second equation? Explain why this operation does not produce an equivalent equation.

If we subtract 5 from both sides of the equation, we get 2x = 8 whose solution is also 4. Although we can subtract 5 from both sides of the equation, we cannot subtract 5 from the expression 2x + 5 alone without changing its value, because clearly 2x + 5 does not equal 2x. Thus, the operations we can use to create an equivalent equation include some that we cannot use to create an equivalent expression. Example 7 (a) Is the equation 2x + 6 = 10 equivalent to the equation x + 3 = 5? (b) Is the expression 2x + 6 equivalent to the expression x + 3?

### Categories by Aristotle

by Donald

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