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AlgebraQuestion and Answers: Page 347

Question Number 13081 Answers: 2 Comments: 0

{ ((x+y+z=[1]_5 )),((xy=[2]_5 )),((yz=[1]_5 )) :} Solve system on Z_5

$$\begin{cases}{{x}+{y}+{z}=\left[\mathrm{1}\right]_{\mathrm{5}} }\\{{xy}=\left[\mathrm{2}\right]_{\mathrm{5}} }\\{{yz}=\left[\mathrm{1}\right]_{\mathrm{5}} }\end{cases} \\ $$$${Solve}\:{system}\:{on}\:\mathbb{Z}_{\mathrm{5}} \\ $$

Question Number 13034 Answers: 0 Comments: 4

MrW1 Before we concluded that: Φ=Σ_(x=0) ^m Σ_(y=0) ^n (1−sgn(x−x′)) If you do: Σ_(x=0) ^1 Σ_(y=0) ^1 (1−sgn(x−x′)) =Σ_(x=0) ^1 Σ_(y=0) ^1 (1−sgn(x−((LCM(x,y))/y))) =(1−sgn(0−((LCM(0,0))/0)))+(1−sgn(1−((LCM(1,0))/0)) +(1−sgn(0−((LCM(0,1))/1)))+(1−sgn(1−((LCM(1,1))/1)) =(1−sgn(−((LCM(0,0))/0)))+(1−sgn(1−(0/0))) +(1−sgn(−(0/1)))+(1−sgn(1−(1/1))) =????

$$\mathrm{MrW1} \\ $$$$\: \\ $$$$\mathrm{Before}\:\mathrm{we}\:\mathrm{concluded}\:\mathrm{that}: \\ $$$$\Phi=\underset{{x}=\mathrm{0}} {\overset{{m}} {\sum}}\underset{{y}=\mathrm{0}} {\overset{{n}} {\sum}}\left(\mathrm{1}−\mathrm{sgn}\left({x}−{x}'\right)\right) \\ $$$$\: \\ $$$$\mathrm{If}\:\mathrm{you}\:\mathrm{do}: \\ $$$$\underset{{x}=\mathrm{0}} {\overset{\mathrm{1}} {\sum}}\underset{{y}=\mathrm{0}} {\overset{\mathrm{1}} {\sum}}\left(\mathrm{1}−\mathrm{sgn}\left({x}−{x}'\right)\right) \\ $$$$=\underset{{x}=\mathrm{0}} {\overset{\mathrm{1}} {\sum}}\underset{{y}=\mathrm{0}} {\overset{\mathrm{1}} {\sum}}\left(\mathrm{1}−\mathrm{sgn}\left({x}−\frac{\mathrm{LCM}\left({x},{y}\right)}{{y}}\right)\right) \\ $$$$=\left(\mathrm{1}−\mathrm{sgn}\left(\mathrm{0}−\frac{\mathrm{LCM}\left(\mathrm{0},\mathrm{0}\right)}{\mathrm{0}}\right)\right)+\left(\mathrm{1}−\mathrm{sgn}\left(\mathrm{1}−\frac{\mathrm{LCM}\left(\mathrm{1},\mathrm{0}\right)}{\mathrm{0}}\right)\right. \\ $$$$+\left(\mathrm{1}−\mathrm{sgn}\left(\mathrm{0}−\frac{\mathrm{LCM}\left(\mathrm{0},\mathrm{1}\right)}{\mathrm{1}}\right)\right)+\left(\mathrm{1}−\mathrm{sgn}\left(\mathrm{1}−\frac{\mathrm{LCM}\left(\mathrm{1},\mathrm{1}\right)}{\mathrm{1}}\right)\right. \\ $$$$\: \\ $$$$=\left(\mathrm{1}−\mathrm{sgn}\left(−\frac{\mathrm{LCM}\left(\mathrm{0},\mathrm{0}\right)}{\mathrm{0}}\right)\right)+\left(\mathrm{1}−\mathrm{sgn}\left(\mathrm{1}−\frac{\mathrm{0}}{\mathrm{0}}\right)\right) \\ $$$$+\left(\mathrm{1}−\mathrm{sgn}\left(−\frac{\mathrm{0}}{\mathrm{1}}\right)\right)+\left(\mathrm{1}−\mathrm{sgn}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{1}}\right)\right) \\ $$$$\: \\ $$$$=???? \\ $$

Question Number 13031 Answers: 1 Comments: 0

MrW1 Going off of Q12883 How many unique angles angles in Z^3 ? What about Z^n ?

$$\mathrm{MrW1} \\ $$$$ \\ $$$$\mathrm{Going}\:\mathrm{off}\:\mathrm{of}\:\mathrm{Q12883} \\ $$$$\: \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{unique}\:\mathrm{angles}\:\mathrm{angles}\:\mathrm{in}\:\mathbb{Z}^{\mathrm{3}} ? \\ $$$$\mathrm{What}\:\mathrm{about}\:\mathbb{Z}^{{n}} ? \\ $$

Question Number 13029 Answers: 1 Comments: 1

Question Number 13002 Answers: 1 Comments: 0

3^((x − 3)(x − y − 2)) = 1 5^((x^2 − 2xy + y^2 + x − y − 3/2)) = (√5) Find the value of x and y

$$\mathrm{3}^{\left({x}\:−\:\mathrm{3}\right)\left({x}\:−\:{y}\:−\:\mathrm{2}\right)} \:=\:\mathrm{1} \\ $$$$\mathrm{5}^{\left({x}^{\mathrm{2}} \:−\:\mathrm{2}{xy}\:+\:{y}^{\mathrm{2}} \:+\:{x}\:−\:{y}\:−\:\mathrm{3}/\mathrm{2}\right)} \:=\:\sqrt{\mathrm{5}} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}\:\mathrm{and}\:{y} \\ $$

Question Number 12927 Answers: 0 Comments: 1

Σcos ((1/n))

$$\Sigma\mathrm{cos}\:\left(\frac{\mathrm{1}}{{n}}\right) \\ $$

Question Number 12909 Answers: 1 Comments: 0

Question Number 12889 Answers: 2 Comments: 1

f(x−1)=(2/3)+f(x) f(0)=36−f(21) ⇒f(0)=?

$${f}\left({x}−\mathrm{1}\right)=\frac{\mathrm{2}}{\mathrm{3}}+{f}\left({x}\right) \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{36}−{f}\left(\mathrm{21}\right) \\ $$$$\Rightarrow{f}\left(\mathrm{0}\right)=? \\ $$

Question Number 12885 Answers: 1 Comments: 0

f(f(x))=f^2 (x) What is a solution?

$${f}\left({f}\left({x}\right)\right)={f}^{\mathrm{2}} \left({x}\right) \\ $$$$\: \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{a}\:\mathrm{solution}? \\ $$

Question Number 12883 Answers: 1 Comments: 0

for −128≤x≤127 and −127≤y≤128 where x,y∈Z Point P(x,y) is a point on the cartesian plane. From the origin, angle θ is made counter −clockwise with the positive x−axis. (1) How many unique angles θ exist if x,y∈P? (2) Furthermore, how many unique angles θ exist for the full range of x,y∈Z?

$$\mathrm{for}\:\:\:\:\:−\mathrm{128}\leqslant{x}\leqslant\mathrm{127} \\ $$$$\mathrm{and}\:\:\:−\mathrm{127}\leqslant{y}\leqslant\mathrm{128} \\ $$$$\mathrm{where}\:\:\:{x},{y}\in\mathbb{Z} \\ $$$$\: \\ $$$$\mathrm{Point}\:{P}\left({x},{y}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{point}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{cartesian}\:\mathrm{plane}. \\ $$$$\: \\ $$$$\mathrm{From}\:\mathrm{the}\:\mathrm{origin},\:\mathrm{angle}\:\theta\:\mathrm{is}\:\mathrm{made}\:\mathrm{counter} \\ $$$$−\mathrm{clockwise}\:\mathrm{with}\:\mathrm{the}\:\mathrm{positive}\:{x}−\mathrm{axis}. \\ $$$$\: \\ $$$$\left(\mathrm{1}\right)\:\mathrm{How}\:\mathrm{many}\:\mathrm{unique}\:\mathrm{angles}\:\theta\:\mathrm{exist} \\ $$$$\mathrm{if}\:{x},{y}\in\mathbb{P}? \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Furthermore},\:\mathrm{how}\:\mathrm{many}\:\mathrm{unique} \\ $$$$\mathrm{angles}\:\theta\:\mathrm{exist}\:\mathrm{for}\:\mathrm{the}\:\mathrm{full}\:\mathrm{range}\:\mathrm{of}\:{x},{y}\in\mathbb{Z}? \\ $$

Question Number 12797 Answers: 1 Comments: 2

1 + x + x^2 + ... x^(49) = (1/2)(x^(49) − (1/x)) Find the value of x

$$\mathrm{1}\:+\:\mathrm{x}\:+\:\mathrm{x}^{\mathrm{2}} \:+\:...\:\mathrm{x}^{\mathrm{49}} \:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{x}^{\mathrm{49}} \:−\:\frac{\mathrm{1}}{\mathrm{x}}\right) \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x} \\ $$

Question Number 12763 Answers: 0 Comments: 0

Solve simultaneously 2x + y − z = 8 ........... equation (i) x^2 − y^2 + 2z^2 = 14 .......... equation (ii) 3x^3 + 4y^3 + z^3 = 195 ........... equation (iii)

$$\mathrm{Solve}\:\mathrm{simultaneously} \\ $$$$\mathrm{2x}\:+\:\mathrm{y}\:−\:\mathrm{z}\:=\:\mathrm{8}\:\:\:\:\:\:\:\:...........\:\mathrm{equation}\:\left(\mathrm{i}\right) \\ $$$$\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{y}^{\mathrm{2}} \:+\:\mathrm{2z}^{\mathrm{2}} \:=\:\mathrm{14}\:\:\:\:\:\:\:..........\:\mathrm{equation}\:\left(\mathrm{ii}\right) \\ $$$$\mathrm{3x}^{\mathrm{3}} \:+\:\mathrm{4y}^{\mathrm{3}} \:+\:\mathrm{z}^{\mathrm{3}} \:=\:\mathrm{195}\:\:\:\:\:\:\:\:\:...........\:\mathrm{equation}\:\left(\mathrm{iii}\right) \\ $$

Question Number 12698 Answers: 0 Comments: 0

Given A^((−1)) = [((0.5 3)),((4 2)) ] find (A^2 )

$$\mathrm{Given}\:\:\:\mathrm{A}^{\left(−\mathrm{1}\right)} =\begin{bmatrix}{\mathrm{0}.\mathrm{5}\:\:\:\:\:\:\mathrm{3}}\\{\mathrm{4}\:\:\:\:\:\:\:\:\:\:\mathrm{2}}\end{bmatrix} \\ $$$$\mathrm{find}\:\left(\mathrm{A}^{\mathrm{2}} \right) \\ $$

Question Number 12682 Answers: 1 Comments: 0