i%2Ba%2B5b%3D15%2B58i.jpeg "(2a-4b)i+a+5b=15+58i")
Solving Complex Equations: A Step-by-Step Guide
This article will guide you through the process of solving the complex equation (2a-4b)i + a + 5b = 15 + 58i.
Understanding 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
To solve this equation, we need to separate the real and imaginary components.
-
Equate Real and Imaginary Parts:
Since the equation involves complex numbers, we need to equate the real and imaginary parts separately. This leads to two equations:
- Real Part: a + 5b = 15
- Imaginary Part: (2a - 4b)i = 58i
-
Solve the Imaginary Part:
We can simplify the imaginary part by canceling out the i on both sides:
- 2a - 4b = 58
-
Solve the System of Equations:
Now we have a system of two linear equations with two unknowns:
- a + 5b = 15
- 2a - 4b = 58
We can solve this system using various methods such as substitution or elimination. Let's use elimination:
-
Multiply the first equation by 2: 2a + 10b = 30
-
Subtract the second equation from this new equation: 14b = -28
-
Solve for b: b = -2
-
Substitute b = -2 into the first equation: a + 5(-2) = 15
-
Solve for a: a = 25
Solution
The solution to the equation (2a-4b)i + a + 5b = 15 + 58i is a = 25 and b = -2.
Verification
We can verify our solution by substituting the values of a and b back into the original equation:
- (2(25)-4(-2))i + 25 + 5(-2) = 15 + 58i
- 58i + 15 = 15 + 58i
This confirms that our solution is correct.
Conclusion
Solving complex equations involves separating the real and imaginary parts and solving the resulting system of equations. The above example demonstrates a step-by-step process to find the solution for a complex equation. Remember, the key is to break down the problem into manageable steps and utilize your knowledge of complex numbers and linear equations.