Math Problem Statement

Find the values of x and y that make the equation true.

5+y+(3x−7)i=9−3i

List the value of x and then the value of y, separated by a comma.

Solution

To solve the equation 5+y+(3x7)i=93i5 + y + (3x - 7)i = 9 - 3i, we can separate the real and imaginary parts and equate them individually.

Step 1: Separate real and imaginary parts.

The given equation is:

(5+y)+(3x7)i=93i(5 + y) + (3x - 7)i = 9 - 3i

  • The real part is 5+y5 + y.
  • The imaginary part is (3x7)i(3x - 7)i.

On the right-hand side:

  • The real part is 99.
  • The imaginary part is 3i-3i.

Step 2: Equate the real parts.

Equating the real parts:

5+y=95 + y = 9

Solve for yy:

y=95=4y = 9 - 5 = 4

Step 3: Equate the imaginary parts.

Equating the imaginary parts:

3x7=33x - 7 = -3

Solve for xx:

3x=3+7=43x = -3 + 7 = 4

x=43x = \frac{4}{3}

Final answer:

The values of xx and yy that make the equation true are:

x=43,y=4x = \frac{4}{3}, \quad y = 4

Thus, xx and yy are 43,4\frac{4}{3}, 4.


Do you want further details or have any questions? Here are 5 related questions to explore:

  1. How can you solve similar equations involving complex numbers?
  2. What is the importance of separating real and imaginary parts in equations?
  3. How do you solve linear equations involving imaginary terms?
  4. Can you explain the properties of imaginary numbers in algebra?
  5. How would the solution change if the imaginary terms were more complex?

Tip: Always separate real and imaginary parts in complex number equations to simplify solving.

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Math Problem Analysis

Mathematical Concepts

Algebra
Complex Numbers
Equating Real and Imaginary Parts

Formulas

Equating real and imaginary parts of a complex number equation

Theorems

Equality of complex numbers

Suitable Grade Level

Grades 9-12