(3+2i)a+(4-i)b=6-7i

3 min read Jun 16, 2024

Solving Complex Equations: (3 + 2i)a + (4 - i)b = 6 - 7i

This article will guide you through solving the complex equation (3 + 2i)a + (4 - i)b = 6 - 7i. We'll break down the steps and explain the concepts involved.

Understanding Complex Numbers

Complex numbers are numbers that extend the real number system by including the imaginary unit, i, where i² = -1. A complex number is typically expressed in the form a + bi, where a and b are real numbers.

Solving the Equation

Expand the equation:

(3 + 2i)a + (4 - i)b = 6 - 7i

3a + 2ai + 4b - bi = 6 - 7i
Group real and imaginary terms:

(3a + 4b) + (2a - b)i = 6 - 7i
Equate real and imaginary components:

For two complex numbers to be equal, their real and imaginary parts must be equal. Therefore:
- 3a + 4b = 6
- 2a - b = -7
Solve the system of equations:

We now have a system of two linear equations with two unknowns. We can solve for a and b using various methods like substitution or elimination.
- Substitution Method:
  - Solve the second equation for b: b = 2a + 7
  - Substitute this value of b into the first equation: 3a + 4(2a + 7) = 6
  - Simplify and solve for a: 3a + 8a + 28 = 6 => 11a = -22 => a = -2
  - Substitute the value of a back into the equation for b: b = 2(-2) + 7 = 3
Solution:

Therefore, the solution to the equation (3 + 2i)a + (4 - i)b = 6 - 7i is a = -2 and b = 3.

Verification

To verify our solution, we can substitute the values of a and b back into the original equation:

(3 + 2i)(-2) + (4 - i)(3) = 6 - 7i

-6 - 4i + 12 - 3i = 6 - 7i

6 - 7i = 6 - 7i

This confirms that our solution is correct.

Conclusion

By applying the principles of complex numbers and solving a system of equations, we have successfully found the solution for the equation (3 + 2i)a + (4 - i)b = 6 - 7i. This problem illustrates the importance of understanding complex number operations and how to manipulate them effectively in solving equations.