Math Problem Statement

Find values for x and y if

3x + yi = 5x + 1 + 2i

Solution

We are given the equation involving complex numbers:

3x+yi=5x+1+2i3x + yi = 5x + 1 + 2i

Step 1: Separate real and imaginary parts

To solve for xx and yy, we separate the real and imaginary components on both sides of the equation.

On the left-hand side:

  • The real part is 3x3x
  • The imaginary part is yy

On the right-hand side:

  • The real part is 5x+15x + 1
  • The imaginary part is 22

Thus, we can split the equation into two separate equations:

  1. Real parts: 3x=5x+13x = 5x + 1
  2. Imaginary parts: y=2y = 2

Step 2: Solve the real part equation

From the first equation 3x=5x+13x = 5x + 1, we solve for xx:

3x5x=13x - 5x = 1 2x=1-2x = 1 x=12x = -\frac{1}{2}

Step 3: Solve the imaginary part equation

From the second equation y=2y = 2, we already know that:

y=2y = 2

Final Solution:

x=12,y=2x = -\frac{1}{2}, \quad y = 2

Would you like more details on the steps or have any other questions?


Here are 5 related questions you might find helpful:

  1. What is a complex number and how do you handle equations involving them?
  2. How do you separate the real and imaginary parts in complex equations?
  3. What is the geometric representation of complex numbers on the complex plane?
  4. Can complex numbers have more than one solution in certain cases?
  5. How do you simplify equations with complex conjugates?

Tip: When working with complex numbers, always remember to handle real and imaginary parts separately to simplify your solution process!

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Algebra
Real and Imaginary Parts

Formulas

Separating real and imaginary parts of complex numbers

Theorems

Equality of complex numbers

Suitable Grade Level

Grades 9-11