Question

What are the values of x and y that satisfy the equation (x + yi)(4 + 9i) = 9 - 4i?

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Answer 1

To solve the equation (x + yi)(4 + 9i) = 9 - 4i, we perform complex multiplication. We then separate the real and imaginary parts, forming two simultaneous linear equations to solve for x and y.

Step-by-step explanation:

To find the values of x and y that satisfy the equation (x + yi)(4 + 9i) = 9 - 4i, we need to perform complex multiplication and compare the real and imaginary parts of the complex numbers.

First, we expand the left side using the distributive property:

(x * 4) + (x * 9i) + (yi * 4) + (yi * 9i) = 9 - 4i

Simplified, it becomes:

4x + 9xi + 4yi - 9y = 9 - 4i
Here, we used the fact that i^2 = -1.

Now we separate the real and imaginary parts:

Real Part: 4x - 9y = 9
Imaginary Part: 9x + 4y = -4 (we drop the i since we are equating the coefficients of i).

To solve for x and y, we can use simultaneous equations:

• 4x - 9y = 9 (multiply by -4)
• 9x + 4y = -4

By solving these equations simultaneously, we can find the correct values of x and y that satisfy the original equation.