281-ա,գ. Լուծեք հավասարումների համակարգը․
ա)
/x + y – 7 = 0
\x2 + xy + y2 = 43
գ)
/x + y = 3
\x² – y² – 4xy + 11 = 0
y = 3 – x
x² – y² – 4xy + 11 = 0
x² – (3 – x)² – 4x(3 – x) + 11 = 0
x² – 9 + 6x – x² – 12x + 4x² + 11 = 0
4x² – 6x + 2 = 0
D = b² – 4ac = (-6)² – 4 * 4 * 2 = 36 – 32 = 4
x = (-b ± √D)/2a = (6 ± √4)/(2 * 4) = (6 ± 2)/8 = 0,75 ± 0,25
x1 = 0,75 + 0,25 = 1
x2 = 0,75 – 0,25 = 0,5
y1 = 3 – x1 = 3 – 1 = 2
y2 = 3 – x2 = 3 – 0,5 = 2,5
Պատ․ (x; y) ∈ {(1; 2), (0,5; 2,5)}
282-ա,գ. Լուծեք հավասարումների համակարգը․
ա)
/x + y = 2
\9x² – 3xy + y = 1
y = 2 – x
9x² – 3xy + y = 1
9x² – 3x(2 – x) + 2 – x = 1
9x² – 6x + 3x² + 2 – x = 1
12x² – 7x + 2 = 1
12x² – 7x + 1 = 0
D = b² – 4ac = (-7)² – 4 * 12 = 49 – 48 = 1
x = (-b ± √D)/2a = (7 ± 1)/24
x1 = (7 + 1)/24 = 8/24 = 1/3
x2 = (7 – 1)/24 = 6/24 = ¼
y1 = 2 – x1 = 6/3 – 1/3 = 5/3
y2 = 2 – x2 = 8/4 – 1/4 = 7/4
Պատ․ (x; y) ∈ {(1/3; 5/3), (¼, 7/4)}
գ)
/2x + y = 1
\3x² = (y – 2)² – 2x
y = 1 – 2x
3x² = (1 – 2x – 2)² – 2x
3x² = (- 2x – 1)² – 2x
3x² = 4x² + 4x + 1 – 2x
x² + 2x + 1 = 0
D = b² – 4ac = 2² – 4 = 4 – 4 = 0
x = (-b ± √D)/2a = -2/2 = -1
y = 1 – 2x = 1 – 2 * (-1) = 1 + 2 = 3
Պատ․ (x; y) = (-1; 3)
283-ա,գ. Լուծեք հավասարումների համակարգը․
ա)
/x + y + z = 6
|y + z = 3
\z = 1
z = 1
/x + y + 1 = 6
\y + 1 = 3
/x + y = 5
\y = 2
y = 2
{x + 2 = 5
{x = 3
Պատ․ (x; y; z) = (3; 2; 1)
գ)
/x + y + z = 2
|x + z = 1
\x + y = 3
/x + y + z = 2
|x + z = 1
\x + y = 3
/x + y + z = 2
|z = 1 – x
\y = 3 – x
{x + 3 – x + 1 – x = 2
{4 – 2x = 2
{2 = 2x
{x = 1
z = 1 – x = 1 – 1 = 0
y = 3 – x = 3 – 1 = 2
Պատ․ (x; y; z) = (1; 2; 0)
284-ա,գ. Լուծեք հավասարումների համակարգը․
ա)
/1/x + 1/y = 1 1/2
\x – 1 = 1
/1/x + 1/y = 1 1/2
\x = 2
{1/2 + 1/y = 1 1/2
{1/y = 1
{y = 1/1 = 1
Պատ․ (x; y) = (2; 1)
գ)
ՏՆԱՅԻՆ․
281-բ,դ. Լուծեք հավասարումների համակարգը․
բ)
/x + y – 6 = 0
\2x² – y² = -23
y = 6 – x
2x² – y² = -23
2x² – (6 – x)² = -23
2x² – 36 + 12x – x² = -23
x² + 12x – 13 = 0
D = b² – 4ac = 12² – 4 * -13 = 144 + 52 = 196
x = (-b ± √D)/2a = (-12 ± √196)/2 = (-12 ± 14)/2 = -6 ± 7
x1 = -6 + 7 = 1
x2 = -6 – 7 = -13
y1 = 6 – x1 = 6 – 1 = 5
y2 = 6 – x2 = 6 + 13 = 19
Պատ․ (x; y) ∈ {(1; 5), (-13, 19)
դ)
/x + y = 12
\2xy = 9(x – y)
/x + y = 12
\2xy = 9(x – y)
y = 12 – x
2xy = 9(x – y)
2x(12 – x) = 9(x – (12 – x))
24x – 2x² = 9(2x – 12)
24x – 2x² = 18x – 108
12x – x² = 9x – 54
-x² + 12x – (9x – 54) = 0
-x² + 12x – 9x + 54 = 0
-x² + 3x + 54 = 0
D = b² – 4ac = 3² – 4 * -1 * 54 = 9 + 216 = 225
x = (-b ± √D)/2a = (-3 ± √225)/(2 * -1) = (-3 ± 15)/-2
x1 = (-3 + 15)/-2 = 12/-2 = -6
x2 = (-3 – 15)/-2 = -18/-2 = 9
y1 = 12 – x1 = 12 + 6 = 18
y2 = 12 – x2 = 12 – 9 = 3
Պատ․ (x; y) ∈ {(-6; 18), (9; 3)}
282-բ,դ. Լուծեք հավասարումների համակարգը․
բ)
/x – 3y = 1
\2xy – x² + 9y² = 11 – 4x
x = 3y + 1
2xy – x² + 9y² = 11 – 4x
2y(3y + 1) – (3y + 1)² + 9y² = 11 – 4(3y + 1)
6y² + 2y – 9y² – 6y – 1 + 9y² = 11 – 12y – 4
6y² + 8y – 8 = 0
3y² + 4y – 4 = 0
D = b² – 4ac = (4)² – 4 * 3 * -4 = 16 + 48 = 64
y = (-b ± √D)/2a = (-4 ± √64)/6 = (-4 ± 8)/6 = (-2 ± 4)/3
y1 = (-2 + 4)/3 = 2/3
y2 = (-2 – 4)/3 = -6/3 = -2
x1 = 3y1 + 1 = 2 + 1 = 3
x2 = 3y2 + 1 = –6 + 1 = -5
Պատ․ (x; y) ∈ {(3; 2/3), (-5; -2)}
դ)
/x – 4y = 10
\(x – 1)² = 7(x + y) + 1
x = 10 + 4y
(10 + 4y – 1)² = 7(10 + 4y + y) + 1
(4y + 9)² = 7(10 + 5y) + 1
16y² + 72y + 81= 71 + 35y
16y² + 37y + 10 = 0
D = b² – 4ac = 37² – 4 * 16 * 10 = 1369 – 640 = 729
y = (-b ± √D)/2a = (-37 ± √729)/32 = (-37 ± 27)/32
y1 = (-37 + 27)/32 = -10/32 = -5/32
y2 = (-37 – 27)/32 = -64/32 = -2
x1 = 10 + 4y1 = 10 + 4 * -5/32 = 320/32 + -20/32 = 300/32 = 75/8
x2 = 10 + 4y2 = 10 + 4 * -2 = 10 – 8 = 2
Պատ․ (x; y) ∈ {(75/8; -5/32), (2; -2)}
283-բ,դ. Լուծեք հավասարումների համակարգը․
բ)
/x + y + z = 0
|x + z = 2
\x = -1
z – 1 = 2
z = 3
x + y + z = 0
y + 3 – 1 = 0
y = -2
Պատ․ (x; y; z) = (-1; -2; 3)
դ)
/x + y + z = 2
|y + z = 3
\x + y = 1
1 + z = 2
z = 1
x + 3 = 2
x = -1
y – 1 = 1
y = 2
Պատ․ (x; y; z) = (-1; 2; 1)
284-բ,դ. Լուծեք հավասարումների համակարգը․
բ)
/1/x – 1/y = –1/4
\y + 1 = 3
y = 2
1/x – 1/2 = –1/4
1/x – 2/4 = –1/4
1/x – 1/4 = 0
1/x = 1/4
x = 4
Պատ․ (x; y) = (4; 2)
դ)
/1/y + 1/x = 5/6
\x + y = 5
y = 5 – x
1/(5 – x) + 1/x = 5/6
x/(5 – x) + 1 = 5x/6
x + 5 – x = 5x(5 – x)/6
5 = 25x – 5x²/6
30 = 25x – 5x²
-5x² + 25x – 30 = 0
x² – 5x + 6 = 0
D = b² – 4ac = (-5)² – 4 * 6 = 25 – 24 = 1
x = (-b ± √D)/2a = (5 ± √1)/-2 = (5 ± 1)/-2 = 2,5 ± 0,5
x1 = 2,5 + 0,5 = 3
x2 = 2,5 – 0,5 = 2
y1 = 5 – x1 = 5 – 3 = 2
y2 = 5 – x2 = 5 – 2 = 3
Պատ. (x; y) ∈ {(3; 2), (2; 3)}
Դավիթ Ապիտոնյան 