Sheet 3. Sequences and Their Limits Exercise 3.3.

Exercise 3.3. Find the limits:

a) ∞ b) −∞ c) 1 d) 2 e) ∞ f) −∞

g) 0 h) 0 i) ²₇ j) 8 k) 0 l) 0 m) −∞ n) 0 o) ¹₄ p) 3 q) ∞ r) 0 s) ⁹₄ t) 0 u) e v) 1 w) ¹₄ x) −¹₂ y) 3 z) 1 aa) ²₃ ab) 0 ac) 0 ad) 0 ae) e⁻³ af) e² ag) e⁸ ah) e⁹ ai) 1

Sheet 4. Limits and Continuity of Functions

Exercise 4.1. Find the limits (if they exists):

a) ¹₂ b) 6 c) −1 d) ⁴₃ e) −₁₆¹ f) ⁵₃ g) ¹₂ h) ¹₂ i) e³ j) 1 k) 0 l) 1 m) ¹₃, n) e o) ∞ p) −¹4 q) ¹₂√

2 r) 2

Exercise 4.2. Compute the one-sided limits of the given functions as x approaches x0: a) lim

x→3⁻

1

x − 3 = −∞, lim

x→3⁺

1

x − 3 = ∞ b) lim

x→3⁻

1

3 − x = ∞, lim

x→3⁺

1

3 − x = −∞

c) lim

x→3⁻

1

(3 − x)² = ∞, lim

x→3⁺

1

(3 − x)² = ∞ d) lim

x→1⁻

x + 1

x − 1 = −∞, lim

x→1⁺

x + 1 x − 1 = ∞ e) lim

x→2⁻

1

x²− 4 = −∞, lim

x→2⁺

1

x²− 4 = ∞ f) lim

x→1⁻2^x−11 = 0, lim

x→1⁺2^x−11 = ∞ g) lim

x→2⁻4^x2−41 = 0, lim

x→2⁺4^x2−41 = ∞ h) lim

x→−2⁻e^4−x21 = 0, lim

x→−2⁺e^4−x21 = ∞ i) lim

x→0⁻

x

1 + e^1x = 0, lim

x→0⁺

x 1 + e^x1 = 0

Sheet 5. Derivatives. L’Hospital’s Rule

Exercise 5.1. Differentiate the following functions:

1) 0 2) 6x + _x¹2 + 4x³+ 1 2√

x 3) 6x²− 2x

4) 13

(2x − 3)² 5) 4x

(x²+ 1)² 6) − 6 x²

x⁶− 2x³+ 1 7) ¡

2x²+ 1¢ 1

√x²+ 1 8) −¹2(x + 1) x 1

x⁵² 9) 2xe

10) e^x¡

x + 3x⁵+ x⁶− 2¢

x³ 11) 10^xln 10 12) 1

4^x − xln 4 4^x 13) x − 3√

x

x³² 14) ln x + 1 15) 2x²ln x − x²− 1

(x²+ 1)²x 16) 1

x ln 3 17) cos x − sin x 18) x³cos x + 3x²sin x

19)

1

2cos x − x sin x

√x 20) − 4x³sin x − x⁴cos x − 4 cos x

(2x + x²+ 2)²(x²− 2x + 2)² 21) 2 sin 2x + 1

22) 0 23) arcsin x + x

√1 − x² 24) 1

x²+ 1+ 1

25) − 1

(2x + x²+ 1)q

1

x+1(1 − x)

26) e^x+ 2e^x√ e^x+ 1 2e^x+ 2e^x√

e^x+ 1 + 2 27) 2xe^x2−3x−4− 3e^x2−3x−4

28) − sin^√√x_x+1⁻¹ 2x +√

x + x³² 29) 30x²¡

2x³− 1¢4

30) 5¡

x²+ 1¢4¡

2x + x²− 1¢ (x + 1)⁶

31) 3 − 3 cos x

2 cos x + cos²x + 1 32) −3 sin 4x − 3 sin 12x 33) −4√

2 − 6x²√ 2 3x⁵√

2x²+ 1 34) 4 × 2^2x+ 4 (ln 2) 2^2x(2x + 1) 35) tan√

x +₂√¹ x(√⁴

x + 1)¡ tan²√

x + 1¢

36) cos 2x + cos 4x

37) 1

2q

1 −¹₄x² 38) − 5

4 (1 − 5x)³⁴p 1 −√

1 − 5x 39) 2

x²+ 1

Exercise 5.2. For the given functions f find f⁰, f⁰⁰, f⁰⁰⁰:

a) ln x+1,_x¹, −x¹² b) cos x−sin x+2x cos x−x sin x−x²sin x, cos x−2 sin x−x cos x−4x sin x−x²cos x, x sin x−5 sin x−

6x cos x−3 cos x+x²sin x c) x

√x²+ 1, 1

√x²+ 1− x²

√x²+ 1 + x²√

x²+ 1, x² (x²+ 1)³

µ x

√x²+ 1+ 2x√

x²+ 1 + x³

√x²+ 1

¶

−

3 x

√x²+ 1 + x²√ x²+ 1

Exercise 5.4. Using L’Hospital’s Rule find the limits:

a) ³₂ b) 0 c) 1

d) ¹₂ e) 1 f) ln e − 1 e − e g) ¹₂ h) 1 i) 1

j) ∞ k) 0 l) 0

m) ¹₂ n) 1 o) 1

Sheet 6. Investigation of Functions

Exercise 6.2. Find the local maximum and minimum values of the following functions:

a) {13, 14} , at {[x = 2] , [x = 3]} b) {−5} , at {[x = −1]} c) ©

−¹4,¹₄ª

, at {[x = −2] , [x = 2]}

d) {−2, 0} , at {[x = −1] , [x = 1]} e) ©

−¹4

ª, at ©£

x = ¹₄¤ª f) Exercise 6.6. Investigate the function f and then sketch its graph:

a)

2.5 1.25 0

-1.25 -2.5

5

2.5

0

-2.5

-5

x y

x y

b)

2.5 1.25 0

-1.25 -2.5

5

2.5

0

-2.5

-5

x y

x y

c)

5 2.5 0

-2.5 -5

10

5

0

-5

-10

x y

x y

d)

5 2.5

0 -2.5

-5

15

10

5

0

-5

x y

x y

e)

1 0.5

0 -0.5

-1

1

0.5

0

-0.5

-1

x y

x y

f)

5 3.75 2.5

1.25 0

1.25

0

-1.25

-2.5

x y

x y

g)

5 3.75 2.5

1.25 0

0

-1.25

-2.5

-3.75

-5

x y

x y

5 2.5

0 -2.5 -5

1.5

1.25

1

0.75

0.5

0.25 0

x y

x y

i)

5 2.5

0 -2.5

-5

15

10

5

0

-5

-10

x y

x y

Sheet 7. Indefinite Integrals

Exercise 7.1. Using basic properties of integral find:

a) ¹₄x⁴− x³+ x² b) ¹₆x⁶−³₄x⁴+³₂x²− ln x c) − x − ln (x − 1) d) ¹₃x³− x + arctan x e) ⁹₅√³

x⁵−2x¹² −⁴5x⁵² f) 3√³

x −₃^√44_x3

g) ₅₆³x^32−16+7x

3 2

√3

x h) _{ln 3}⁴ √⁴

3^x i) ₁₅⁸

q xp

x³²x j) − ⁵_{ln 5}^−x −²_{ln 2}^−x k) −_e^1x − 2 ln (e^x) l) e^x+¹₂e^2x+ ln (e^x)

m) sin x − cos x n) − cos¹2x sin¹₂x +¹₂x o) − cot x +¹2π − arccot (cot x) p) sin x cos x¹ −_{sin x}² cos x

Exercise 7.2. Using the change of variable method (substitution method) find:

a) ¹₂ln¡ x²+ 1¢

b) −_10(x2¹+3)⁵ c) ¹₃arctan¡ e^3x¢

d) ¡

−1 + x²¢ _x²₋₂

√−(−1+x²)³

e) ²₅(x − 3)⁵² + 2 (x − 3)³² f) ²₉(3x + 1)³² g) ¹₃¡

x²+ 1¢³₂

h) − e^1x i) −¹₂√

2 arctan√^√2

x²−2 j) √

x²− 9 k) ¹₄arcsin x⁴ l) −¹₂√

4 + e^−4x m) −¹2ln (3 + 2 cos x) n) −¹2cos²x o) sin (ln x) p) −¹2e^−x2

Exercise 7.3. Find using the integration by parts:

a) cos x + x sin x b) x²e^x− 2xe^x+ 2e^x

c) ¹₂e^xcos x +¹₂e^xsin x d) − ¹2x cos²x +¹₄sin x cos x +¹₄x e) ¹₂x²ln²x −¹₂x²ln x +¹₄x² f) − ^{ln x}_x −¹_x

g) − x cot x + ln (sin x) h) ^e_x^x

i) − x²cos x + 2 cos x + 2x sin x j) − ¹5e^2xcos x +²₅e^2xsin x k) x tan x + ln (cos x) l) −¹3xe^−3x−¹9e^−3x Exercise 7.4. Find:

a) ¹₂sin x cos x +¹₂x b) −¹2sin x cos x +¹₂x c) ¹₆sin⁶x d) 2√

1 + sin x e) ¹₂¡

x²+ 1¢ ln¡

x²+ 1¢

−¹2 −¹2x² f) ²₃(2 + ln |x|)³² g) ¹₂arctan x² h) ¹₃arcsin x³

Sheet 8. Definite Integrals

Exercise 8.1. Compute the integrals:

a) ¹₈π b) ¹₃π c) − 2e⁻¹+ 1 d) − 2π e) ⁴₃ f) ¹₄π −¹₂ln 2 g) ¹₂ h) ¹₂

Sheet 9. Improper Integrals. The Gamma and Beta Functions

Exercise 9.1. Compute the integral:

a) 1 b)√

5 c) ∞ d) ¹₂π e) ¹₉π f) ¹₃ g) ¹₂ h) ¹₂ i) ∞ j) −¹2

k) 2. 583 9 l) 0 m) undefined n) Exercise 9.2. Compute:

a) 4 b) ¹⁵₈√

π c) 720

d) ₃₆¹√

3π e) ₅₁₂¹⁵ f) ₂₇¹π√ 3 ^√34

Γ(²3) g) ₁₆¹π h) ¹₈π i) ₂₃₁₀¹ j) _{13 135 122}¹ k) 3814 697 265 625

111 384 l) ₅₀₄¹

Sheet 10. Matrices

Exercise 10.1.

a) A + B =

"

1 5 3 3

# , 2A =

"

6 2 0 −4

#

, 2A − B =

"

8 −2

−3 −9

#

, A − αB =

"

3 + 2α 1 − 4α

−3α −2 − 5α

#

b) A+B =

⎡

⎢⎣

−2 8 4

1 7 8

−1 ⁷2 5

⎤

⎥⎦ , 2A =

⎡

⎢⎣

0 14 2

2 0 8

−4 1 6

⎤

⎥⎦ , 2A−B =

⎡

⎢⎣

2 13 −1

2 −7 4

−5 −2 4

⎤

⎥⎦ , A−αB =

⎡

⎢⎣

2α 7 − α 1 − 3α

1 −7α 4 − 4α

−2 − α ¹2 − 3α 3 − 2α

⎤

⎥⎦

c) A+B =

"

3 −1 √

2 4 9 −2 −√

3

# , 2A =

"

2 −6 0

6 8 −4

#

, 2A−B =

"

0 −8 −√ 2 5 3 −4 +√

3

#

, A−αB =

"

1 − 2α −3 − 2α − 3 − α 4 − 5α −2

"

0 8 + α # "

2α 2 − 2α # "

3α −5 − 4α # "

α + α² 1 − α − α (7 + 2α) #

Exercise 10.2.

a)

"

1 −3 0

3 4 −2

#⎡

⎢⎣

−2 1 3 0 7 4 1 3 2

⎤

⎥⎦ =

"

−2 −20 −9

−8 25 21

# ,

"

1 −3 0

3 4 −2

#⎡

⎢⎣

−2 0 1 1 7 3 3 4 2

⎤

⎥⎦ =

"

−5 −21 −8

−8 20 11

#

, niemo˙zliwe, niemo˙zliwe,

b)

"

3 1

0 −2

# "

−2 4 3 5

#

=

"

−3 17

−6 −10

# ,

"

3 1

0 −2

# "

−2 3 4 5

#

=

"

−2 14

−8 −10

# ,

"

−2 4 3 5

# "

3 1

0 −2

#

=

"

−6 −10 9 −7

# ,

"

−2 4 3 5

# "

3 0 1 −2

#

=

"

−2 −8 14 −10

#

Exercise 10.3.

a) X =

"

3 −5 2 1 −11 3

#

− 2

"

1 −4 −2

3 7 0

#

=

"

1 3 6

−5 −25 3

# ,

b) X =

"

20 −4

−2 −5

# , Y =

"

−13 4

3 3

#

Exercise 10.4.

a)

¯¯

¯¯

¯

7 0

1 −1

¯¯

¯¯

¯= −7 b)

¯¯

¯¯

¯¯

¯

1 3 −2

2 4 5

−1 0 −2

¯¯

¯¯

¯¯

¯

= −19 c)

¯¯

¯¯

¯¯

¯¯

¯¯

1 0 1 7

0 −3 2 2 0 0 4 −4

0 0 0 5

¯¯

¯¯

¯¯

¯¯

¯¯

= −60

Exercise 10.5.

a)

¯¯

¯¯

¯¯

¯¯

¯¯

−1 −2 4 1

−3 1 3 1

0 2 1 4

1 −1 2 0

¯¯

¯¯

¯¯

¯¯

¯¯

= 65 b)

¯¯

¯¯

¯¯

¯¯

¯¯

0 1 3 −2

1 2 −1 4

−1 −3 −5 0

1 3 −2 1

¯¯

¯¯

¯¯

¯¯

¯¯

= −11

Exercise 10.6.

a)

¯¯

¯¯

¯¯

¯¯

¯¯

1 −2 0 0

0 1 3 5

2 4 −1 −1

−2 2 0 1

¯¯

¯¯

¯¯

¯¯

¯¯

= −29,

b)

¯¯

¯¯

¯¯

¯¯

¯¯

¯¯

2 −1 3 1 3

0 1 0 −1 0

1 3 0 2 −1

3 −1 −1 −2 −1

0 3 1 2 1

¯¯

¯¯

¯¯

¯¯

¯¯

¯¯

= 49

c)

¯¯

¯¯

¯¯

¯¯

¯¯

a 3 0 1

0 b 2 −1

1 0 c 3

−5 d 0 1

¯¯

¯¯

¯¯

¯¯

¯¯

= abc + 6ad + adc + 96 − 2d + 15c + 5bc

Exercise 10.7.

a) x = −6, x = −4 b) x = 0 Exercise 10.8.

a)

"

−5 3 2 −1

# b)

⎡

⎢⎣

2 3

1 3 −¹3

−¹₃ ¹₃ −¹₃

−¹3 −²3 5 3

⎤

⎥⎦ c)

⎡

⎢⎣

1 2 3 0 1 5 1 2 3

⎤

⎥⎦ d)

⎡

⎢⎢

⎢⎢

⎣

1

2 −2 0 0 0 0 ¹₂ 0

1

2 −2 0 1

0 1 0 0

⎤

⎥⎥

⎥⎥

⎦ ,

Exercise 10.9. b) X =

"

−¹⁷32 −32⁹ 11 32

19 32

#

Exercise 10.10. Reduce to the base form:

a)

⎡

⎢⎣

1 2 5 0 0 0 0 0 0

⎤

⎥⎦ b)

⎡

⎢⎣

1 0 ¹₄ ⁵₄ ³₄ 0 1 ¹₄ −³4 −¹4

0 0 0 0 0

⎤

⎥⎦ c)

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎣

1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎦

d)

⎡

⎢⎢

⎢⎢

⎣

1 0 −⁴₇ 0 0 1 ¹³₇ 0

0 0 0 1

0 0 0 0

⎤

⎥⎥

⎥⎥

⎦

Sheet 11. Systems of Linear Equations

Exercise 11.1. Solve the following systems:

(a) by means of Cramer’s Rule, (b) by means of elementary operations.

a)

⎧⎪

⎨

⎪⎩ x₁= 1 x2= 0 x₃= −1

, b)

⎧⎪

⎨

⎪⎩ x₁= 1 x2= ¹₂ x₃= −2

, c)

⎧⎪

⎨

⎪⎩

x₁= 0 x2= 0 x₃= 0

, d)

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ x₁= 0 x2= ₁₃⁸ x3= 0 x₄= −₁₃²

, e)

⎧⎪

⎨

⎪⎩ x = 2 y = −1 z = −3

, f)

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ t = 1 x = 4 y = −2 z = 0

, g)

⎧⎪

⎨

⎪⎩

x₁= −1 x2= 1 x₃= 2

,

h)

⎧⎪

⎨

⎪⎩ x₁= 0 x2= 1 x₃= −1

, i)

⎧⎪

⎨

⎪⎩ x₁= 3 x2= 2 x₃= 1

, j)

⎧⎪

⎨

⎪⎩

x₁= −1 x2= 0 x₃= 1 Exercise 11.2. Solve the following systems:

a)

⎧⎪

⎨

⎪⎩

x1= x3+¹⁸₃₁ x₂= x₃−₃₁¹ x3∈ R

, b)

⎧⎪

⎨

⎪⎩

x1= −x³ x₂= 2x₃ x3∈ R

, c)

⎧⎪

⎨

⎪⎩

x1= −²⁷7

x₂= −₁₄⁹ x3= −¹⁸7

, d) inconsistent (no solutions)

Exercise 11.3. Find the general solutions of the following systems. Indicate two distinct particular solutions:

a)

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪

x₁= −²5x₃+¹₅x₄ x2= ¹₅x3−³5x4

x3∈ R

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪

x₁= 0 x2= 0 x3= 0

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪

x₁= −¹5

x2= −²5

x3= 1 , b)

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪

x₁= −⁷3x₃ x2= ⁵₃x3

x3∈ R

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪

x₁= −⁷3

x2=⁵₃ x3= 1

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪

x₁= −7 x2= 5 x3= 3

c)

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

x1= −2x³− 2x⁴ x2= −x⁴ x₃∈ R x4∈ R

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

x1= −2 x2= 0 x₃= 1 x4= 0

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

x1= −2 x2= −1 x₃= 0 x4= 1

, d)

⎧⎪

⎨

⎪⎩

x1= −3x³− 5 x₂= −x3

x3∈ R

⎧⎪

⎨

⎪⎩

x1= −5 x₂= 0 x3= 0

⎧⎪

⎨

⎪⎩

x1= −8 x₂= −1 x3= 1

Exercise 11.4. Find one of the basis solutions of the following systems:

a)

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

x1= ⁴₃ x2= ⁷₃ x₃= 0 x4= 0

, b)

⎧⎪

⎨

⎪⎩ x1= 0 x₂= −¹₂ x3= 0

, c)

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ x1= 1 x2= 1 x₃= 0 x4= 0

, d)

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

x₁=¹²₇ x2=⁹₇ x₃=¹¹₇ x4= 0 x5= 0 Exercise 11.5. Find all basis solutions of the systems:

a)

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ x1= 0 x2= 0 x₃=⁷₃ x4=²₃

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ x1= 0 x2=¹₂ x₃= 0 x4= 1

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ x1= 0 x2= −1 x₃= 7 x4= 0

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

x1= −1 x2= 0 x₃= 0 x4= 1

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ x1= 2 x2= 0 x₃= 7 x4= 0

, b)

⎧⎪

⎨

⎪⎩

x1=³₄ x₂=¹₄ x3= 0

⎧⎪

⎨

⎪⎩ x1=⁴₅ x₂= 0 x3=¹₅

⎧⎪

⎨

⎪⎩ x1= 0 x₂= −⁷₂ x3= −3

,

c)

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

x1= −¹¹5

x₂= −³₅ x3=²⁹₅ x4=³³₅ x₅= 0

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

x1= ¹¹₂ x₂= ¹₂ x3= −3 x4= 0 x₅= ¹¹₂

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

x1=²³₈ x₂=¹₈ x3= 0 x4=⁹₄ x₅=²⁹₈

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩ x1= 2 x₂= 0 x3= 1 x4= 3 x₅= 3

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩ x1= 0 x₂= −²₇ x3=²³₇ x4=³³₇ x₅=¹¹₇

, d)

⎧⎪

⎨

⎪⎩

x₁= ³₄ x2= ⁹₄ x3= 0

⎧⎪

⎨

⎪⎩ x₁= 0 x2= ³₂ x3= −3

⎧⎪

⎨

⎪⎩

x₁=₁₀³ x2= 0 x3= −⁹5

Exercise 11.6. Solve the system of equations by means of elementary operations on rows. What are the basis variables and the free variables of the found general solution? Indicate two distinct particular solutions of the system, one of them should be the basis solution. Find the conditions for the general solution to be nonnegative

a)

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

t = −x − 1 x ∈ R y = 1 z = 0

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

t = −1 x = 0 y = 1 z = 0

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ t = 0 x = −1 y = 1 z = 0

the system has not nonnegative solution,

b)

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

t = z + 1 x = −2z + 6 y = z + 2 z ∈ R

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ t = 1 x = 6 y = 2 z = 0

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

t = −1 x = 10 y = 0 z = −2

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ t ≥ 0 x ≥ 0 y ≥ 0 z ≥ 0

⇔ z ∈ [0, 3]

Exercise 11.7. (a)

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

x₁= ²₃x₃−²3x₄ x2= 1 − x³+ x4

x3∈ R x₄∈ R

, (b) impossible, (c)

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

x₁∈ R x2= 1 −³2x1

x3= ³₂x1+ x4

x₄∈ R