$$ 2 x+2 y+z-7=0,2 x-y+3 z-3=0,4 x+5 y-2 z-12=0 $$ tekisliklarning kesishish nuqtasi va $$ M(0 ; 3 ; 0), N(1: 1 ; 1) $$ nuqtalardan o'tuvchi tekislik tenglamasi topilsin.

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Uchta tekislikning tenglamalari quyidagicha berilgan:
$$
\begin{cases}
2x + 2y + z = 7 \quad \text{(1)} \
2x - y + 3z = 3 \quad \text{(2)} \
4x + 5y - 2z = 12 \quad \text{(3)}
\end{cases}
$$

**Kesishish nuqtasini topish:**

1. **(1) va (2) tenglamalarni yechish:**

$$
   \begin{align*}
   (2x - y + 3z) - (2x + 2y + z) &= 3 - 7 \
   -3y + 2z &= -4 \quad \text{(4)}
   \end{align*}
   $$

2. **(3) tenglamani (1) tenglamaning ikki baravari bilan kamaytirish:**

$$
   \begin{align*}
   (4x + 5y - 2z) - 2(2x + 2y + z) &= 12 - 14 \
   y - 4z &= -2 \quad \text{(5)}
   \end{align*}
   $$

3. **(5) dan y ni ifodalash va (4) ga qo'yish:**

$$
   y = 4z - 2
   $$
   $$
   -3(4z - 2) + 2z = -4 \
   -12z + 6 + 2z = -4 \
   -10z = -10 \
   z = 1
   $$
   $$
   y = 4(1) - 2 = 2
   $$

4. **(1) tenglamadan x ni topish:**

$$
   2x + 2(2) + 1 = 7 \
   2x + 5 = 7 \
   2x = 2 \
   x = 1
   $$

Demak, uchta tekislikning kesishish nuqtasi $$ (1, 2, 1) $$ hisoblanadi.

**Tekislik tenglamasini topish:**

Berilgan nuqtalar:
- $$ M(0, 3, 0) $$
- $$ N(1, 1, 1) $$
- Kesishish nuqtasi $$ (1, 2, 1) $$

1. **Vektorlarni aniqlash:**
   $$
   \vec{A} = M - (1, 2, 1) = (-1, 1, -1)
   $$
   $$
   \vec{B} = N - (1, 2, 1) = (0, -1, 0)
   $$

2. **Normal vektorni topish (kross mahsuloti):**
   $$
   \vec{n} = \vec{A} \times \vec{B} = (-1, 1, -1) \times (0, -1, 0) = (-1)\mathbf{i} + 0\mathbf{j} + 1\mathbf{k} = (-1, 0, 1)
   $$

3. **Tekislik tenglamasi:**
   $$
   -1(x - 1) + 0(y - 2) + 1(z - 1) = 0 \
   -x + z = 0 \
   x - z = 0
   $$

Demak, $$ M $$ va $$ N $$ nuqtalardan o'tuvchi tekislik tenglamasi:
$$
x - z = 0
$$
Kesishish nuqtasi $$ (1, 2, 1) $$. M va N nuqtalardan o'tuvchi tekislik tenglamasi $$ x - z = 0 $$.

tekisliklarning kesishish nuqtasi
va nuqtalardan o’tuvchi tekislik tenglamasi topilsin.

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tekisliklarning kesishish nuqtasi va nuqtalardan o’tuvchi tekislik tenglamasi topilsin.