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Equation Solving
Brackets is the second in a series of fifteen puzzle caches of increasing difficulty based on solving equations. It is recommended that one start with Abacus (GC6A006) and continue in alphabetical sequence through Brackets (GC6AR81), Compass (GC6AWD0), Decagon (GC6C4EE), Ellipse (GC6CFW3), FOIL (GC6HDNE), Googol (GC751JK), Hectare (GC75BD3), Index (GC75EAM), Jargon (GC75X8W), Kite (GC76010), LessThan (GC763HV ), Median (GC7DH17), Nonagon (GC7DWC4) and Obelus (GC7DWCD).
The Name - Brackets
The "B" in "BEDMAS" stands for "brackets." "BEDMAS" is a memory device by which the order of operations in simplifying mathematical expressions is summarized and we’ll learn more about that in future caches. It is sufficient for what we are doing today to remember that, when simplifying mathematical expressions, the rule is that any possible simplification within the brackets is done before anything else. If there are two sets of “nesting” brackets - [(--------) + (--------)] - we work from the inner brackets outwards. But more of that later; the point is that brackets in mathematics are very important and hence worthy of being the name of a cache . . . and we needed a name starting with a “B!”
Simplification of Mathematical Expressions
“Expression” and “Term”
Simply put, an “expression” in mathematics is a “string” of items (called “terms”) which are separated by plus or minus signs. For example, 3 + 4x - 7c2 + 17 - 2p + y3 is an “expression” of six items or “terms.”
A single “term” may have several numbers and/or letters but they are united by multiplication or division signs. For example: 4x means “4 X x” so it is a term; 7c2 means “7 X c X c” so it is a term; 3y/2 means 3 X y divided by 2 so it is a term.
”Numerical Coefficient” and “Literal Coefficient”
A term may have both a ”numerical coefficient” (a number) and a ”literal coefficient” (a letter). For example, the term “3" is simply a “numerical coefficient” and y3 is a “literal coefficient.” However, 7c2 has both a “numerical coefficient” that being 7 and a “literal coefficient” that being c2. Note that exponents (like the "2" in this case) are not numerical coefficients!
”Like” Terms and “Unlike” Terms
”Like” terms have the same literal coefficients and can be added together.
For example, 3x2 and 5x2 are like terms so can be added to yield 8x2 in the same way that 3 bananas and 5 bananas can be added together and expressed as 8 bananas.
However, 4a2 and 7a3 are unlike terms and, if added together still remain as 4a2 + 7a3 in the same way that 4 apples and 3 pears remain as 4 apples and 3 pears (unless you express them as 1 bowl of fruit!)
Smplifying Expressions
To simplify an expression, one scans for a certain “like term” checking each one off as it is added to the total and then repeats for each “like term” in the expressions until there are none left. We say we “add” because we see 8x - 3x as “8x plus negative 3x” . . . which amounts to the same thing really but in some cases makes the process easier to understand. Note: x means 1x.
Examples of Simplifying Expressions
(1)
4x + 5 + 7x - 2
= 11x + 3.
(2)
5x + 1 - 2y + (4x + 9 + 7y - 2x - 4y) . . . note that we're doing the work in the brackets first!
= 5x + 1 - 2y + (2x + 3y + 9)
= 5x + 1 - 2y + 2x + 3y + 9
= 7x + y + 10
(3)
7d - 5 + (3d - 1 + 2d) + (5d + 4 - 8d)
= 7d - 5 + (5d - 1) + (- 3d + 4)
= 7d - 5 + 5d - 1 - 3d + 4
= 9d - 2
(4)
3x - 2x2 + 1 + 7x + 5 x2 - 4
=10x + 3x2 - 3
Now the Puzzle!
Simplify each expression and identify the requested number:
(1)
4x - 5x + 3x
= __________ .
A is the numerical coefficient of the x term: _____ .
(2)
5a - 2 + 3a + 7 + 2a - 1
= _________ .
B is the ones digit of the numerical coefficient of the a term: _____ .
(3)
9a + 7b + 3c + 2b - 5c + 2a
= _______________________________ .
C is the numerical coefficient of the b term: _____ .
(4)
3x2 - 2x + 5 + 4x2 + 2x - 3
= ______________________________________ .
D is the numerical coefficient of the x term: _____ .
G is the term that has no literal coefficient: _____ .
K is the numerical coefficient of the x2 term: _____ .
(5)
6x3 - 9x2 + 8x - 2 - 6x + 5x2 + 7 + 3x3
= ________________________ .
E is the numerical coefficieint of the x3 term: _____ .
F is the numerical coefficient of the x2 term: _____ .
(6)
7p2 + 5q3 - 8t + 3t + 4q3 - 5p2
= ___________________________ .
H is the numerical coefficient of the q3 term: _____ .
J is the numerical coefficient of the p2 term: _____ .
The Co-ordinates of the Cache
You are looking for
N 44° A B . C D E’ and W 078° F G . H J K’
which you now know (be careful in transposing) to be
N 44° __ __ . __ __ __ ‘ and W 078° __ __ . __ __ __ ‘.
Comments
- Please bring your own writing instrument.
- You can check your answers for this puzzle on GeoChecker.com.