(4-5i)x+(2+3i)y=10-7i

Jasonbradley

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Jun 16, 2024

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Solving Complex Equations: (4-5i)x + (2+3i)y = 10-7i

This article will guide you through the process of solving a complex equation involving variables. We'll focus on the equation (4-5i)x + (2+3i)y = 10-7i.

Understanding Complex Numbers

Before we dive into the solution, let's clarify a few key points about complex numbers:

• Complex Numbers: They consist of two parts: a real part and an imaginary part. They are typically written in the form a + bi, where 'a' is the real part and 'b' is the imaginary part. 'i' represents the imaginary unit, where i² = -1.

• Operations on Complex Numbers: Addition, subtraction, multiplication, and division can be performed on complex numbers, following specific rules.

Solving the Equation

Our goal is to find the values of x and y that satisfy the given equation. To achieve this, we'll use the following steps:

1. Separate the Real and Imaginary Parts: We'll group the terms with the real parts and the terms with the imaginary parts.

   • Real part: (4x + 2y)
   • Imaginary part: (-5x + 3y)i

2. Equate the Real and Imaginary Parts: For the equation to hold true, both the real and imaginary parts on both sides must be equal.

   • 4x + 2y = 10
   • -5x + 3y = -7

3. Solve the System of Equations: Now we have a system of two linear equations with two unknowns. We can solve this using methods like:

   • Substitution: Solve one equation for one variable and substitute it into the other equation.
   • Elimination: Multiply one or both equations by constants to make the coefficients of one variable the same, then add or subtract the equations to eliminate that variable.

4. Obtain the Values of x and y: After solving the system of equations, we'll find the values of x and y that satisfy the original complex equation.

Example Solution

Let's use the elimination method to solve the system of equations:

1. Multiply the first equation by 5 and the second equation by 4:

   • 20x + 10y = 50
   • -20x + 12y = -28

2. Add the two equations:

   • 22y = 22

3. Solve for y:

   • y = 1

4. Substitute y = 1 into either of the original equations and solve for x:

   • 4x + 2(1) = 10
   • 4x = 8
   • x = 2

Therefore, the solution to the equation (4-5i)x + (2+3i)y = 10-7i is x = 2 and y = 1.

Conclusion

Solving complex equations involving variables requires a systematic approach. By separating the real and imaginary parts, forming a system of equations, and utilizing methods like elimination or substitution, we can find the values of the variables that satisfy the given equation.