Question

1) x-8i+(y-3)i=1 2) (3+i)x-2(1+4i)y=-2-4 найдите действительные числа x и y изз урав.

Answer 1

Комплексные числа подчиняются всем тем же аксиомам сложения(вычитания) и умножения(деления), что и действительные числа.

Просто аккуратно раскрываем скобки и приводим подобные

Обозначим действительную и комплексную части выражения слева как $\Re_1$ и $\Im_1$. Соответственно из выражения справа - $\Re_2$ и $\Im_2$

$1) x-8i+(y-3)i=1 \medskip \\ x-8i+yi-3i-1=0 \medskip \\ (x-1)-i(11-y)=0 \medskip \\ \begin{cases} \Re_1=\Re_2\\\Im_1=\Im_2\end{cases} \medskip \\ \begin{cases}x-1=0\\11-y=0\end{cases}\Leftrightarrow\begin{cases}x=1\\y=11\end{cases}$

$2) x(3+i)-2y(1+4i)=-2-4 \medskip \\ 3x+xi-2y-8yi=-6 \medskip \\ (3x-2y)+i(x-8y)=-6 \medskip \\ \begin{cases}\Re_1=\Re_2\\\Im_1=\Im_2\end{cases} \medskip \\ \begin{cases}3x-2y=-6\mid \cdot 4\\x-8y=0\end{cases} \medskip \\ a)-\begin{cases}12x-8y=-24\\x-8y=0\end{cases} \medskip \\ 11x=-24 \medskip \\ x=-\dfrac{24}{11} \medskip \\ b) x-8y=0 \medskip \\ -\dfrac{24}{11}=8y \medskip \\ y=-\dfrac{24}{88}=-\dfrac{3}{11} \medskip \\ \begin{cases}x=-\dfrac{24}{11}\medskip\\y=-\dfrac{3}{11}\end{cases}$

Answer 2

To find the real values of x and y from the given equations, we need to equate the real and imaginary parts of both equations.

Let's start with equation (1):

1. x - 8i + (y - 3)i = 1

Equating real parts:

x = 1

Equating imaginary parts:

-8 + (y - 3) = 0 y - 3 = 8 y = 8 + 3 y = 11

So, from equation (1), we have x = 1 and y = 11.

Now, let's move on to equation (2):

2. (3 + i)x - 2(1 + 4i)y = -2 - 4

Expanding the equation:

3x + ix - 2y - 8iy = -6

Now, separate the real and imaginary parts:

Real Part: 3x - 2y = -6 ... (Equation A) Imaginary Part: ix - 8iy = 0

Since we know that x = 1 from the previous equation, we can substitute it into the imaginary part:

1 - 8iy = 0

Solving for y:

-8iy = -1 y = 1/8

So, from equation (2), we have x = 1 and y = 1/8.

In conclusion, the solutions for the real numbers x and y are x = 1 and y = 11 (from equation 1) and x = 1 and y = 1/8 (from equation 2).