Thanks I appreciate help!
One thing I'm trying to understand is why X + 0.16X = 1.16X.
I believe it has something to do with X + X = 2X.
I can't seem to figure out how to apply it to any other problem though.
Again I appreciate the help!
This is actually a common hang up with beginning algebra students.
Two cookies plus three cookies equals five cookies. But that rule does not apply just to cookies. 2 of some thing plus 3 of that thing equals 5 of that thing. And it applies to collections of things. 2 boxes of 10 cookies each plus 3 boxes of 10 cookies each = 5 boxes of 10 cookies each. Of course, we can then proceed to say that represents 50 cookies because all the numbers are known. In algebra, however, all the numbers are not known, at least not initially.
We
algebraically summarize the idea that adding collections of things is no different than adding individual things as
\(\displaystyle ab + ac \equiv a(b + c).\)
That is the distributive law of multiplication over addition. It is usually justified to students by numeric examples such as
\(\displaystyle 3 *7 + 3 * 5 = 21 + 15 = 36 = 3(12) = 3(7 + 5) \implies 3 * 7 + 3 * 5 = 3(7 + 5).\)
The way algebra is taught today, students are made to
MEMORIZE that formula's
NAME, but they are seldom explicitly told its crucial importance in simplifying algebraic expressions. (In fact, no kid I have tutored remembers
ever having been told anything about why he or she was told to memorize the commutative and associative laws and the distributive law. I hate the way math is being taught in US high schools.)
In algebra, we come across expressions like
\(\displaystyle 7x + 5x\). Using the commutative law of multiplication twice, then the distributive law, and finally the commutative law again, we get
\(\displaystyle 7x + 5x = x * 7 + 5x = x * 7 + x * 5 = x(7 + 5) = x * 12 = 12x.\)
In practice, we leave out the intermediate steps and simplify
\(\displaystyle 7x + 5x = 12x.\)
What a student with more than three neurons remembers as a practical rule is to add up the coefficients of like variables when simplifying.
\(\displaystyle 4a + 0.2a - 3a + 7.1a = (4 + 0.2 - 3a + 7.1)a = 8.3a.\)
4 meters plus 2 tenths of a meter - 3 meters + 7.1 meters = 8.3 meters. Basic. The formula using a says that is true whether we are counting meters, pounds, inches, students, or collections.
But the student may then be baffled by \(\displaystyle x + 0.16x.\)
The first term does not appear to have a coefficient. But what does x stand for? A number. And every number when multiplied by 1 equals itself.
\(\displaystyle x \equiv 1 * x \equiv 1x.\) So
\(\displaystyle x + 0.16x = 1x + 0.16x = (1 + 0.16)x = 1.16x.\)
When someone forgets that x = 1x, that person is then stumped when told x + 0.16x = 1.16x. The same person would have no trouble at all figuring out that
\(\displaystyle 3 + 3 * 0.16 = 3 * 1 + 3 * 0.16 = 3(1 + 0.16) = 3 * 1.16 = 3.48.\)