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(a) To handle the system (5) of four equations, Cramer told us that "we will find the value of each unknown by forming n fractions of which the common denominator has as many terms as there are diverse arrangements of n different things." In this case, n = 4, so how many terms will the "common denominator" expression in each of the four formulas contain?

(a) To handle the system (5) of four equations, Cramer told us that "we will find the value of each unknown by forming n fractions of which the common denominator has as many terms as there are diverse arrangements of n different things." In this case, n = 4, so how many terms will the "common denominator" expression in each of the four formulas contain?

Algebra: Structure And Method, Book 1
Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter9: Systems Of Linear Equations
Section9.7: Puzzle Problems
Problem 8OE
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Task 8
Task 8 (a) To handle the system (5) of four equations, Cramer told us that "we will find the value of each unknown by forming n fractions of which the common denominator has as many terms as there are diverse arrangements of n different things." In this case, n = 4, so how many terms will the "common denominator" expression in each of the four formulas contain? (b) "Each term is composed of the letters ZYXV," according to Cramer, “to which we distribute, as exponents, the first n numbers arranged in all possible ways." (Again, it is important to note that these exponents are NOT powers, but superscript labels that identify the equation from which the coefficient comes.) Make a list of all these terms (the first of which will be Z'Y²X³V+). Then, by counting derangements in the sequence of exponents that appears in each term, decide which sign (+ or -) is to be assigned to each. Combine all these terms to produce the "common denominator" expression. (c) Finally, follow Cramer's instructions in the final paragraph of the above source text to determine the numerator, and thus the fractional formula for one of the unknowns 2, y, 2, and v of the general system (5). (Choose your unknown at random, via tosses of a pair of coins: if you toss two heads, choose z; if first a head then a tail, choose y; if first a tail then a head, choose z; and if two tails, choose v. Working with just one of these unknowns is tedious enough!)
Transcribed Image Text:Task 8 (a) To handle the system (5) of four equations, Cramer told us that "we will find the value of each unknown by forming n fractions of which the common denominator has as many terms as there are diverse arrangements of n different things." In this case, n = 4, so how many terms will the "common denominator" expression in each of the four formulas contain? (b) "Each term is composed of the letters ZYXV," according to Cramer, “to which we distribute, as exponents, the first n numbers arranged in all possible ways." (Again, it is important to note that these exponents are NOT powers, but superscript labels that identify the equation from which the coefficient comes.) Make a list of all these terms (the first of which will be Z'Y²X³V+). Then, by counting derangements in the sequence of exponents that appears in each term, decide which sign (+ or -) is to be assigned to each. Combine all these terms to produce the "common denominator" expression. (c) Finally, follow Cramer's instructions in the final paragraph of the above source text to determine the numerator, and thus the fractional formula for one of the unknowns 2, y, 2, and v of the general system (5). (Choose your unknown at random, via tosses of a pair of coins: if you toss two heads, choose z; if first a head then a tail, choose y; if first a tail then a head, choose z; and if two tails, choose v. Working with just one of these unknowns is tedious enough!)
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