(2+3i)a+(1-2i)b=4+13i

4 min read Jun 16, 2024

Solving Complex Equations: (2 + 3i)a + (1 - 2i)b = 4 + 13i

This article will guide you through solving the complex equation (2 + 3i)a + (1 - 2i)b = 4 + 13i. We will break down the steps involved and explain the concepts behind them.

Understanding Complex Numbers

Before we begin, let's refresh our understanding of complex numbers. A complex number is a number that can be expressed in 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

Our goal is to find the values of a and b that satisfy the given equation:

(2 + 3i)a + (1 - 2i)b = 4 + 13i

Expand the equation:

We start by distributing the complex numbers:

(2a + 3ai) + (b - 2bi) = 4 + 13i
Combine real and imaginary terms:

Group the real terms and the imaginary terms separately:

(2a + b) + (3a - 2b)i = 4 + 13i
Equate real and imaginary coefficients:

For two complex numbers to be equal, their real and imaginary parts must be equal. Therefore, we obtain two separate equations:
- 2a + b = 4
- 3a - 2b = 13
Solve the system of equations:

Now, we have a system of two linear equations with two unknowns. We can solve this system using various methods like substitution, elimination, or matrix methods.

Using the elimination method:
- Multiply the first equation by 2: 4a + 2b = 8
- Add the second equation to the modified first equation: 7a = 21
- Solve for a: a = 3
- Substitute the value of a back into either of the original equations to solve for b. Using the first equation: 2(3) + b = 4
- Solve for b: b = -2
Solution:

Therefore, the solution to the equation (2 + 3i)a + (1 - 2i)b = 4 + 13i is a = 3 and b = -2.

Verification

We can verify our solution by substituting the values of a and b back into the original equation:

(2 + 3i)(3) + (1 - 2i)(-2) = 4 + 13i

(6 + 9i) + (-2 + 4i) = 4 + 13i

4 + 13i = 4 + 13i

The equation holds true, confirming our solution.

Conclusion

This example demonstrates the process of solving equations involving complex numbers. By breaking down the problem into separate equations for the real and imaginary parts, we can simplify the problem and solve for the unknown variables. This approach is applicable to various complex number equations and can be used to solve a wide range of problems in mathematics, physics, and engineering.