(3+2i)x+2(1-i)y=17-2i

3 min read Jun 16, 2024

Solving Complex Equations: (3+2i)x + 2(1-i)y = 17-2i

This article will guide you through solving the complex equation (3+2i)x + 2(1-i)y = 17-2i. We'll break down the steps to find the values of x and y that satisfy this equation.

Understanding Complex Numbers

Before we dive into the solution, let's remember what complex numbers are. A complex number is a number of the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.

Solving the Equation

Expand the equation: Begin by distributing the coefficients on both sides of the equation:
```
(3+2i)x + 2(1-i)y = 17-2i 
3x + 2ix + 2y - 2iy = 17 - 2i 
```
Separate real and imaginary terms: Rearrange the equation to group the real terms and imaginary terms together:
```
(3x + 2y) + (2x - 2y)i = 17 - 2i
```
Equate coefficients: For two complex numbers to be equal, their real parts and imaginary parts must be equal. This gives us two equations:
```
3x + 2y = 17 
2x - 2y = -2 
```
Solve the system of equations: We now have a system of two linear equations in two unknowns. You can solve this system using various methods like substitution, elimination, or matrices. Let's use elimination here:
- Add the two equations together: 5x = 15
- Solve for x: x = 3
- Substitute x = 3 into either of the original equations to find y. Let's use the first equation: 3(3) + 2y = 17
- Solve for y: y = 4

Solution

Therefore, the solution to the equation (3+2i)x + 2(1-i)y = 17-2i is x = 3 and y = 4.

Verification

You can always check your solution by substituting the values of x and y back into the original equation.

(3 + 2i)(3) + 2(1 - i)(4) = 17 - 2i
9 + 6i + 8 - 8i = 17 - 2i
17 - 2i = 17 - 2i

This confirms that our solution is correct.