Հանրահաշիվ

N 281.
ա) y=7-x
x²+xy+y²=43
x²-xy+y²=43
x²+x(7-x)+(7-x)²=43
x²+7x-x²+49-14x+x²=43
x²-7x+6 = 0
D=49-24=25
x=3,5±2,5
y₁=7-x₁=7-6= 1 x₁=3,5+2,5=6
y₂=7-x₂=7-1= 6 x₂=3,5-2,5=1

բ) y = 6–x
2x²–y²=-23
2x²–(6–x)²=-23
2x²–36+12x–x²=-23
x²+12x–13 = 0
D=144+52 = 196
x = (-12 ± 14)/2=-6 ± 7
x₁ = -6+7 = 1 y₁ = 6–x₁ = 5
x₂ = -6–7 = -13 y₂ = 6–x₂ = 19

գ) 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=36–32 = 4
x = 0,75 ± 0,25
x₁ = 0,75+0,25 = 1 y₁ = 3–x₁=3–1=2
x₂ = 0,75–0,25 = 0,5 y₂ = 3–x₂=3–0,5=2,5

դ) 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+3x+54 = 0
D=9+216 = 225
x = (-3 ± 15)/-2
x₁ = 12–x₁ = 12+6 = 18 y₁ = 12–x₁ = 12+6 = 18
x₂ = (-3–15)/-2 = -18/-2 = 9 y₂ = 12–x₂ = 12–9 = 3:

N 282.
ա) 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 = 49–48 = 1
x = (7 ± 1)/24
x₁ = (7 + 1)/24 = 8/24 = 1/3 y₁ = 2–x₁ = 6/3–1/3 = 5/3
x₂ = (7–1)/24 = 6/24 = 1/4 y₂ = 2–x₂ = 8/4–1/4 = 7/4

բ) 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=16+48 = 64
y = (-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

գ) y = 1–2x
3x²= (1–2x–2)²–2x
3x²= (-2x–1)²–2x
3x²=4x²+4x+1–2x
x²+2x+1 = 0
D = 2²–4 = 4–4 = 0
x =-2/2 =-1
y = 1–2x = 1+2 = 3

դ) 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 = 1369–640 = 729
y = (-37±√729)/32 = (-37 ± 27)/32
y₁ = (-37+27)/32 = -10/32 = -5/32
y₂ = (-37–27)/32 = -64/32 = -2
x₁ = 320/32+-20/32 = 300/32 = 75/8
x₂ = 10+4y₂ = 10–8 = 2:

N 283.
ա) z = 1
x+y+1 = 6 = > x+y = 5
y+1 = 3 = > y = 2
x+2 = 5
x = 3
x; y; z = 3; 2; 1

բ) z–1 = 2
z = 3
x+y+z = 0
y+3–1 = 0
y = -2
x; y; z = -1; -2; 3

գ) x+y+z = 2 => x+y+z = 2
x+z = 1 = > z = 1–x
x+y = 3 = > 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

դ) 1+z = 2
z = 1
x+3 = 2
x = -1
y–1 = 1
y = 2
x; y; z = -1; 2; 1: