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zzzzz

Misol 3.

x _x _x _
3 15 35
x _x
63 99
_x
143
 12
tenglamani yeching.

solve((x/3+x/15+x/35+x/63+x/99+x/143)=12,x);

Misol 4.

³^x^² ²^x^³ 2,5x  2  0
tenglamani yeching.

4 2

solve(((3*x-2)/4+(2*x+3)/2-2.5*x+2)=0,x);

Agar tenglama bir necha yechimga ega bo`lsa, name buyrug’i orqali boshqa hisoblash amallarini bajarishda yechimlarni tanlab olamiz. Berilgan tenglamaning k- yechimini aniqlash uchun kvadrat qavslar ichida yechim tartibini ko`rsatish lozim: name[k]. Masalan:

x:=solve(x^2-a=0,x);

x :  a ,

x[1];



x[2];

x[1]+x[2];

0
Misol 1. Tenglamani yeching.

1998x²  2000x  2  0

solve(1998*x^2-2000*x+2=0,x);

1, ¹
999

x:=solve(1998*x^2-2000*x+2=0,x);

x := 1, ¹
999

x[1];

x[2];

1
1

999

Misol.

x₁va
^xsonlari 3x²  2x  6  0
tenglamaning ildizlari bo’lsa, ildizlari

2
yig’indisi va ayirmasini toping.

x:=solve(3*x^2-2*x-6=0,x);

x[1];

x := ¹+ ¹
3 3
19 , ¹- ¹
3 3

1 ₊1
3 3

x[2];

1 _-1
3 3

x[1]+x[2];

2
3

x[1]-x[2];

2
3

Tenglamalar sistemasini yechish
Tenglamalar sistemasi solve({eq1,eq2,…},{x1,x2,…}),buyrug’I yordamida yechiladi, faqat aylana qavslar ichidagi 1- figurali qavs ichida tenglamalar, ikkinchi figurali qavs ichida esa tenglamaning o`zgaruvchilari kiritiladi. Agar sizga tenglamaning yechimlari bilan bog’liq ravishda keyingi hisoblashlar kerak bo`lsa, solve komandasi name ning qandaydir nomini ifodalaydi. So`ngra assign(name) buyrug’i uni to`ldiradi. Shundan keyin yechimlar ustida matematik amallar bajarish mumkin. Masalan:

s:=solve({a*x-y=1,5*x+a*y=1},{x,y});

s:={ x ^a^¹, y ^a^⁵}
5  a²5  a²

assign(s); simplify(x-y);

6
5  a ²


 ^x^^y_²^y_⁵
Misol 1. Tenglamalar sistemasini yeching ^{2 3 2}

Maple dasturida yechish:



3x


^^ 2 y  0
 2

s:=solve({(x+y)/2-2*y/3=5/2, 3*x/2+2*y=0},{x,y});

s := {y = -3, x = 4}

Javob: (4, -3)
Misol 2. (x,y) sonlar jufti toping.

Maple dasturida yechish:

²^x^^y^⁵


_3x  2 y  4
sistemaning yechimi bo’lsa, x – y ni

s:=solve({2*x-y=5,3*x+2*y=4},{x,y});

s := {x = 2, y = -1}

assign(s);simplify(x-y);

Javob: x – y =3

³^x^^y^⁴⁵


Misol 3. Agar ^x  3y  15 bo’lsa, x+y+z nimaga teng.


^3z  x  6

Maple dasturida yechish:

s:=solve({3*x+y=45, z+3*y=-15, 3*z+x=6},{x,y,z});

s := {z = (-24)/7, y = (-27)/7, x = 114/7} yoki
s : ^z  ^²⁴, y  ^²⁷, x  ¹¹⁴^
 

7

7

7
 

assign(s);simplify(x+y+z);

Javob: x + y + z = 9

Chiziqli va ikkinchi darajali tenglamalar sistemasining yechimini Maple tizimida osongina topish imkoniyati mavjud. Quyidagi Maple tizimida bajarilgan misollar yechimi bilan birgalikda keltirilgan.

Misol 1. Sistemadan
x  y ni toping
^x ²  y ²  xy  8


^x  y  3

Maple dasturida yechish:

s:=solve({x*x+y*y+x*y=8, x+y=3},{x,y});

s := {y = RootOf (_Z ²- 3 _Z + 1, label = _L1), x = -RootOf (_Z ²- 3 _Z + 1, label = _L1) + 3}

assign(s); simplify(x*y);

1

Javob: x  y =1

Misol 2. Sistemaning yechimini toping.

Maple dasturida yechish:

^x ²  y ²  2xy  1


^x  y  3

s:=solve({x^2+y^2-2*x*y=1, x+y=3},{x,y});

s := {y = 1, x = 2}, {y = 2, x = 1}

s[1];

s[2];

Javob: (2;1) va (1;2)
{y = 1, x = 2}

{y = 2, x = 1}

Misol 3. Ushbu
^x^^y^³


^x ²  y ²  6

tenglamalar sistemasidan x ni toping.

Maple dasturida yechish:

s:=solve({x^2-y^2=6, x+y=3},{x,y});

s : ^y  ¹, x  ⁵^
 

2
 ²

assign(s); simplify(x);

5
2

Javob: x=2,5

Misol 4. Tenglamalar sistemasini yeching.

Maple dasturida yechish:

 ^y^⁴^²


^x ² y  2

s:=solve({y+4=2, (x^2)*y=-2},{x,y});

s := {y = -2, x = 1}, {y = -2, x = -1}

s[1];

s[2];

Javob: (-1; -2), (1; -2)
{y = -2, x = 1}

{y = -2, x = -1}

Misol 4. Agar
x  y  5 va
xy  7
bo’lsa,
x³ y  xy ³
ning qiymati qancha bo’ladi?

Maple dasturida yechish:

s:=solve({x-y=5, x*y=7},{x,y});

s := {y = RootOf (_Z ²+ 5 _Z - 7, label = _L1), x = RootOf (_Z ²+ 5 _Z - 7, label = _L1) + 5}

assign(s); simplify(x^3*y+x*y^3);

Javob:
x³ y  xy ³ =273
273

Misol 5. Agar
a  b  12
va  ab  a²  144 bo’lsa, a ning qiymati qanchaga teng?

Maple dasturida yechish:

k:=solve({a-b=12, (-a)*b+a^2=144}, {a,b});

k := {a = 12, b = 0}

assign(k); simplify(a);

Javob: a=12 Misol 6. Agar toping.
12

x²  4xy  y²  4  2xy ^va

x  y 12

bo’lsa, xy ning qiymatini

Maple dasturida yechish:

k:=solve({x^2-4*x*y+y^2=4-2*x*y, x+y=12}, {x,y});

k := {y = 5, x = 7}, {y = 7, x = 5}

assign(k); simplify(x*y);

Javob: xy =35
Misol 7. b  a  18

va a²  b²  170 ,

ab  ?

Maple dasturida yechish:

t:=solve({b+a=18, a^2+b^2=170}, {a,b});

t := {b = 7, a = 11}, {b = 11, a = 7}

assign(t); simplify(a*b);

Javob: ab  77


 ^xy_¹⁰
^x  y 7

Misol 8. ^^yz ⁴⁰

tenglamalar sistemasidan x ni toping.


^y  z 13


^zx _5
^x  z 8

Maple dasturida yechish:

S:=solve({x*y/(x+y)=10/7, y*z/(y+z)=40/13, z*x/(x+z)=5/8}, {x,y,z});




S : ^z  ⁸⁰, y  ^⁸⁰, x  ⁸⁰^
^49 23 79^

assign(S); simplify(x);

80
79

Javob:
_x_80
79

Tenglamalarning sonli yechimi

Tenglamani sonli yechishda, berilgan transcendent tenglama analitik yechim bermasa, maxsus fsolve(eq,x) buyrug’idan foydalaniladi. Parametr xuddi solve dagi kabi ko`rsatiladi. Masalan:

x:=fsolve(cos(x)=x,x);

x:=.7390851332

§5. Tengsizliklar va tengsizliklar sistemasini yechish.

Oddiy tengsizliklarni yechish

solve buyrug’i tengsizliklarni yechishda ham qo`llaniladi.Tengsizlikning yechimi o`zgaruvchining o`zgarish oralig’i bo`lgan interval ko`rinishida beriladigan tengsizlikning yechimi yarim o`qlarda bo`lsa, u/h RealRange(–, Open(a)), ya’ni x(–, a), a – ixtiyoriy son. Open so`zi interval ochiq chegara degan ma’noni anglatadi. Agar ushbu so`z bo`lmasa, tenglamalr to`plamida bu interval yopiqligini anglatadi.Masalan:

s:=solve(sqrt(x+3)

convert(s,radical);

^ ² ^

^RealRange^^Open
3
²¹^,^

 ^^
Agar siz x(a, b) ko`rinishda ko`rishni istamasangiz, berilagan o`zgaruvchini
a<x, x< b tipda bo`lsa, figurali qavslarda ko`rsatish kerak.Masalan:

> solve(1-1/2*ln(x)>2,{x});

{0  x, x  e⁽^²⁾}

Tengsizliklar sistemasini yechish

solve buyrug’i yordamida tengsizliklar sistemasini ham yechish mumkin.
Masalan:

> solve({x+y>=2,x-2y<=1,x-y>=0,x-2y>=1},{x,y});

{x  1  2 y, ¹ y}
3
Misol 1. Tengsizliklar sistemasi nechta butun yechimga ega?


³^⁴^x^⁵
_2x  3(x 1)  1
Maple dasturida yechish:

> solve({3+4x>=5, 2x-3*(x-1)>-1},x);

^¹ x, x  4^
 

2
 
Javob: Tengsizliklar sistemasi 3 ta butun yechimga ega.

Misol 2. Tengsizliklar sistemasini yeching.

Maple dasturida yechish:

^x(x  1)  10  (x  1)²  3

_3x  4(x  7)  16  3x

> solve({x(x+1)+10>(x+1)^2+3, 3x-4(x-7)>=16- 3x},x);

 6  x, x  6


 ^y^⁵_²^y^³
Misol 3. Tengsizliklar sistemasini yeching. ^{4 3}


^4 y  1

y  4

Maple dasturida yechish:

^2 3

solve({(y-5)/4<(2*y+3)/3, (4*y+1)/2<(y-4)/3},y);

5




^²⁷__y_,_y_^¹¹
 ₁₀

Misol 4. Ushbu yechimlarini toping.
1296 : 314  9x  32  2976 : 96
tengsizlikning barcha natural

Maple dasturida yechish:

solve({1256/314<9*x-32, 9*x-32<=2976/96},x);

4  x, x  7
Javob: Tengsizlikning natural yechimlari 5,6,7
Misol 5. Tengsizliklar sistemasining eng katta butun yechimini ko’rsating
^^x^⁵ 2x  0
 ₄


^2x  8

Maple dasturida yechish:
^x 
 ⁵
 1  2x

> solve({(x+5)/4-2x>=0, x-(2x-8)/5>=1-2*x},x);



7


^^³ x, x  ⁵^
 ₁₃

Javob: Tengsizliklar sistemasining eng katta butun yechimi 0.

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O`zbekiston respublikasi aloqa, axborotlashtirish va telekommunikatsiya texnologiyalari davlat ko`mitasi

Misol 3.

solve((x/3+x/15+x/35+x/63+x/99+x/143)=12,x);

Misol 4.

solve(((3*x-2)/4+(2*x+3)/2-2.5*x+2)=0,x);

x:=solve(x^2-a=0,x);

x[1];

x[2];

x[1]+x[2];

solve(1998*x^2-2000*x+2=0,x);

x:=solve(1998*x^2-2000*x+2=0,x);

x[1];

x[2];

Misol.

x:=solve(3*x^2-2*x-6=0,x);

x[1];

x[2];

x[1]+x[2];

x[1]-x[2];

s:=solve({a*x-y=1,5*x+a*y=1},{x,y});

assign(s); simplify(x-y);

Maple dasturida yechish:

s:=solve({(x+y)/2-2*y/3=5/2, 3*x/2+2*y=0},{x,y});

Maple dasturida yechish:

s:=solve({2*x-y=5,3*x+2*y=4},{x,y});

assign(s);simplify(x-y);

Maple dasturida yechish:

s:=solve({3*x+y=45, z+3*y=-15, 3*z+x=6},{x,y,z});

assign(s);simplify(x+y+z);

Maple dasturida yechish:

s:=solve({x*x+y*y+x*y=8, x+y=3},{x,y});

assign(s); simplify(x*y);

1

Maple dasturida yechish:

s:=solve({x^2+y^2-2*x*y=1, x+y=3},{x,y});

s[1];

s[2];

Maple dasturida yechish:

s:=solve({x^2-y^2=6, x+y=3},{x,y});

assign(s); simplify(x);

Maple dasturida yechish:

s:=solve({y+4=2, (x^2)*y=-2},{x,y});

s[1];

s[2];

Maple dasturida yechish:

s:=solve({x-y=5, x*y=7},{x,y});

assign(s); simplify(x^3*y+x*y^3);

Maple dasturida yechish:

k:=solve({a-b=12, (-a)*b+a^2=144}, {a,b});

assign(k); simplify(a);

Maple dasturida yechish:

k:=solve({x^2-4*x*y+y^2=4-2*x*y, x+y=12}, {x,y});

assign(k); simplify(x*y);

Maple dasturida yechish:

t:=solve({b+a=18, a^2+b^2=170}, {a,b});

assign(t); simplify(a*b);

Maple dasturida yechish:

S:=solve({x*y/(x+y)=10/7, y*z/(y+z)=40/13, z*x/(x+z)=5/8}, {x,y,z});

assign(S); simplify(x);

Tenglamalarning sonli yechimi

x:=fsolve(cos(x)=x,x);

§5. Tengsizliklar va tengsizliklar sistemasini yechish.

Oddiy tengsizliklarni yechish

s:=solve(sqrt(x+3) convert(s,radical);

convert(s,radical);

> solve(1-1/2*ln(x)>2,{x});

Tengsizliklar sistemasini yechish

> solve({x+y>=2,x-2*y<=1,x-y>=0,x-2*y>=1},{x,y});

> solve({3+4*x>=5, 2*x-3*(x-1)>-1},x);

> solve({x*(x+1)+10>(x+1)^2+3, 3*x-4*(x-7)>=16- 3*x},x);

solve({(y-5)/4<(2*y+3)/3, (4*y+1)/2<(y-4)/3},y);

solve({1256/314<9*x-32, 9*x-32<=2976/96},x);

> solve({(x+5)/4-2*x>=0, x-(2*x-8)/5>=1-2*x},x);

solve(((3x-2)/4+(2x+3)/2-2.5*x+2)=0,x);

solve(1998x^2-2000x+2=0,x);

x:=solve(1998x^2-2000x+2=0,x);

x:=solve(3x^2-2x-6=0,x);

s:=solve({ax-y=1,5x+a*y=1},{x,y});

s:=solve({(x+y)/2-2y/3=5/2, 3x/2+2*y=0},{x,y});

s:=solve({2x-y=5,3x+2*y=4},{x,y});

s:=solve({3x+y=45, z+3y=-15, 3*z+x=6},{x,y,z});

s:=solve({xx+yy+x*y=8, x+y=3},{x,y});

s:=solve({x^2+y^2-2xy=1, x+y=3},{x,y});

assign(s); simplify(x^3y+xy^3);

k:=solve({x^2-4xy+y^2=4-2xy, x+y=12}, {x,y});

S:=solve({xy/(x+y)=10/7, yz/(y+z)=40/13, z*x/(x+z)=5/8}, {x,y,z});

s:=solve(sqrt(x+3)

convert(s,radical);

> solve({x+y>=2,x-2y<=1,x-y>=0,x-2y>=1},{x,y});

> solve({3+4x>=5, 2x-3*(x-1)>-1},x);

> solve({x(x+1)+10>(x+1)^2+3, 3x-4(x-7)>=16- 3x},x);

solve({(y-5)/4<(2y+3)/3, (4y+1)/2<(y-4)/3},y);

solve({1256/314<9x-32, 9x-32<=2976/96},x);

> solve({(x+5)/4-2x>=0, x-(2x-8)/5>=1-2*x},x);