Question

\(\frac{x+t y+2+3 t}{2 x+2 i y-3}=i+2\)

Short answer

y = \frac{x}{3}, x = \frac{-9t-24}{t-9}

Step-by-step solution

Isolate the complex number

We want to simplify the given equation by isolating the complex term. Start by moving 2 to isolate the fraction containing real and complex parts: \[\frac{x + t y + 2 + 3 t}{2 x + 2 i y - 3} = i + 2\] Subtract 2 from both sides of the equation: \[\frac{x + t y + 2 + 3 t}{2 x + 2 i y - 3} - 2 = i\]

Simplify the equation

Combine the terms on the left side under a common denominator: \[\frac{x + t y + 2 + 3 t - 2(2 x + 2 i y - 3)}{2 x + 2 i y - 3} = i\]Simplify the numerator:\[x + t y + 2 + 3 t - 4 x - 4 i y + 6 = -3 x + t y - 4 i y + 3 t + 8\]

Set real and imaginary parts equal

For the complex equation to hold, both the real and imaginary parts must be equal.Equate real parts:\[-3 x + t y + 3 t + 8 = 0\]Equate imaginary parts:\[-4 y i = i(2 x + 2 y)\]Thus,\[-4 y = 2 x + 2 y\]

Solve for variables

First solve the imaginary parts equation accurately:\[-4 y = 2 x + 2 y\]Add 4 y on both sides:\[6 y = 2 x\]Therefore,\[y = \frac{x}{3}\]Now solve the real parts equation:\[-3 x + t y + 3 t + 8 = 0\]Substitute the value of y:\[-3 x + t \frac{x}{3} + 3 t + 8 = 0\]\[-3 x + \frac{tx}{3} + 3 t + 8 = 0\]Multiply by 3 to clear the fraction:\[-9 x + tx + 9 t + 24 = 0\]\[(t - 9)x = -9t - 24\]Thus,\[x = \frac{-9t -24}{t - 9}\]

Key concepts

complex numbers

Complex numbers are numbers that have both a real part and an imaginary part. The standard form of a complex number is written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The imaginary unit \(i\) is defined by the property that \(i^2 = -1\). This means that the square of an imaginary number is negative.
For example:

• \(3 + 4i\): Here, 3 is the real part, and 4i is the imaginary part.
• \(1 - 2i\): Here, 1 is the real part, and -2i is the imaginary part.

solving equations

When solving equations that involve complex numbers, we often need to break them down into their real and imaginary components. This allows us to handle each part separately and make the calculations simpler. Let’s take our original equation step by step: \(\frac{x + ty + 2 + 3t}{2x + 2iy - 3} = i + 2\). By isolating the complex term and moving 2 over to the right-hand side, we simplify our problem.
Then, it's key to combine like terms. By converting all complex terms under a common denominator, we arrive at a simpler format. Finally, solving for the variables \(x\) and \(y\) requires isolating both the real and imaginary parts. Separate the equation based on these parts to solve accurately.

real and imaginary parts

In any complex equation, both the real and imaginary parts must be equal for the equation to hold. When we isolate the complex term, we get: \[\frac{x + ty + 2 + 3t}{2x + 2iy - 3} = i + 2\] Next, subtract 2 from both sides to separate the real and imaginary parts: \[\frac{x + ty + 2 + 3t - 2(2x + 2iy - 3)}{2x + 2iy - 3} = i\]
We then equate the real and imaginary parts:
\[-3 x + ty + 3t + 8 = 0 \] (Real Part)
\[-4iy = i(2x + 2iy) \] (Imaginary Part)

algebraic manipulation

Algebraic manipulation involves rearranging equations to isolate and solve for specific variables. Let’s look at how it's done in our example. First, solve the imaginary part: \[-4yi = i(2x + 2iy)\]
Add 4y to both sides:
\[6y = 2x\]
Thus,
\[y = \frac{x}{3}\]
Next, solve the real part by substituting \(y\):

• Start with: \[-3x + t\frac{x}{3} + 3t + 8 = 0\]
  Clear the fraction by multiplying by 3: \[-9x + tx + 9t + 24 = 0\]
  Finally isolate \(x\): \[(t - 9)x = -9t - 24\]
  \[x = \frac{-9t - 24}{t - 9}\]

This step-by-step manipulation lets us solve complex equations efficiently.