An expression is shown below: 3(m + 5 + 9m) part a: write two expressions that are equivalent to the given expression. part b: show that one of your expressions in part a is equivalent to the given expression using algebraic properties. explain which properties you used. part c: show that your other expression from part a is equivalent to the given expression by substituting a number for m.

A) The expression can be simplified, expanded, or factored as you wish.
.. = 3(10m +5)
.. = 15(2m +1)
.. = 30m +15
or, it can be written in any number of fairly useless ways.
.. = m +29m +1 +14
.. = m +2 + 28m +13 + m

B) For all of them, the distributive property, associative properties of addition and multiplication, commutative properties of addition and multiplication are used. The precise list of properties and the order of their use depends on the precise sequence of rewrites you choose (and how detailed you want to get).
.. 3(m +9m +5) . . . . . commutative property of addition (swap 5, 9m)
.. 3((1 +9)m +5) . . . . . distributive property (possibly with commutative property of multiplication. The distributive property is usually written a(b+c) = ab+ac. We have the operand order swapped: ba +ca = (b+c)a.)
.. 3(10m +5) . . . . . . . properties of numbers: 1+9=10; substitution property (10 substituted for 1+9) of equality

C) Let m=7
Original: 3(7+5+9*7) = 3(7+5+63) = 3(12+63) = 3(75) = 225
Rewrite: 3(10*7 +5) = 3(70 +5) = 3(75) = 225
225 = 225 . . . . so the rewrite works for m=7.

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