We are tasked with solving the equation:

$(x + 2y) + (3x - 2y)i = 14 + 10i$

This is a complex number equation, where we can equate the real and imaginary parts separately to find $x$ and $y$.

Step 1: Separate real and imaginary parts

We rewrite the left-hand side by separating the real and imaginary parts:

• The real part is $(x + 2y)$.
• The imaginary part is $(3x - 2y)$.

So, the equation becomes:

$(x + 2y) + (3x - 2y)i = 14 + 10i$

Now, equate the real and imaginary parts on both sides:

• Real part: $x + 2y = 14$
• Imaginary part: $3x - 2y = 10$

Step 2: Solve the system of equations

We now solve the system of linear equations:

1. $x + 2y = 14$
2. $3x - 2y = 10$

Add the two equations:

$(x + 2y) + (3x - 2y) = 14 + 10$

Simplifying:

$4x = 24 \implies x = 6$

Substitute $x = 6$ into the first equation:

$6 + 2y = 14$

Solving for $y$:

$2y = 8 \implies y = 4$

Final Answer:

The solution is the ordered pair $(x, y) = (6, 4)$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you solve systems of linear equations using substitution?
2. Can we solve complex number equations geometrically?
3. How are real and imaginary parts used in solving complex number equations?
4. What happens if the right-hand side is a purely real number?
5. How do we extend this approach to systems involving more variables?

Tip: When solving complex number equations, always treat real and imaginary parts separately!