Question

solve this problem please iv

Answer 1

$\sf\underline{ \large{Explanation:}}$

$\sf 2x + i^9y(2 + i) = xi^7 + 10i^{16}$

first find the exact values of $\sf i^9$ , $\sf i^7$ , $\sf i^{16 }$

$\sf i^9 = i \times i^8 = i \times (i^2 )^4$

$\longrightarrow\sf i \times ( -1)^4 = i$

$\sf i^7 = i \times i^6 = i \times (i^2)^3$

$\longrightarrow\sf i \times (-1)^3$

$\longrightarrow\sf - i$

$\sf i^{16} = (i^2)^8 = (-1)^8 = 1$

therefore,

$\sf 2x + i^9y ( 2 + i) = xi^7 + 10i^{16 }$

$\sf 2x + iy (2 + i) = x (-i) + (10 \times 1 )$

$\sf 2x + 2yi + yi^2 = -xi + 10$

$\sf 2x + 2yi - y + xi = 10$

$\sf 2x - y + xi + 2yi = 10 + 0i$

$\sf (2x - y) + i (x + 2y) = 10 + 0i$

compare the real and imaginary values

$\sf 2x - y = 10$ ------------(1)

$\sf x + 2y = 0$

multiply $\sf x + 2y = 0$ by 2

$\longrightarrow\sf 2x + 4y = 0$ -----------(2)

subtract equation (1) from equation (2)

2x + 4y = 0

2x - y = 10

(-)....(+)....(-)

_____________

5y = -10

$\longrightarrow \sf y = \frac{-10}{5}$

$\large{\sf y=-2}$

put the value of y in equation (1)

$\sf 2x - y = 10$

$\longrightarrow\sf 2x -(-2)=10$

$\longrightarrow\sf 2x + 2 = 10$

$\longrightarrow\sf 2x = 10 - 2$

$\longrightarrow\sf 2x = 8$

$\longrightarrow\sf x = \frac{8}{2}$

$\longrightarrow\sf x = 4$

$\large{\sf x=4}$

therefore the values of x and y are 4 and -2 .