2. a. Find $$ p $$ and $$ q $$ if $$ p+i q=5+3 i $$ b. Find $$ x $$ and $$ y $$ if $$ (3 y+5)+(2 x-5) i=0 $$ c. Find the real numbers $$ x $$ and $$ y $$ if $$ 5 x+3 y+2 i x-3 i y=4+3 i $$

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UpStudy AI · Accepted Answer

Let's solve each part step by step.

### Part a
We need to find $$ p $$ and $$ q $$ such that:
$$
p + i q = 5 + 3 i
$$

To do this, we can equate the real and imaginary parts:
- The real part gives us: 
  $$
  p = 5
  $$
- The imaginary part gives us:
  $$
  q = 3
  $$

Thus, the values are:
$$
p = 5, \quad q = 3
$$

### Part b
We need to find $$ x $$ and $$ y $$ such that:
$$
(3y + 5) + (2x - 5)i = 0
$$

For this equation to hold, both the real part and the imaginary part must equal zero:
1. Real part:
   $$
   3y + 5 = 0
   $$
   Solving for $$ y $$:
   $$
   3y = -5 \implies y = -\frac{5}{3}
   $$

2. Imaginary part:
   $$
   2x - 5 = 0
   $$
   Solving for $$ x $$:
   $$
   2x = 5 \implies x = \frac{5}{2}
   $$

Thus, the values are:
$$
x = \frac{5}{2}, \quad y = -\frac{5}{3}
$$

### Part c
We need to find the real numbers $$ x $$ and $$ y $$ such that:
$$
5x + 3y + 2ix - 3iy = 4 + 3i
$$

Again, we equate the real and imaginary parts:
1. Real part:
   $$
   5x + 3y = 4
   $$
2. Imaginary part:
   $$
   2x - 3y = 3
   $$

Now we have a system of equations:
1. $$ 5x + 3y = 4 $$  (Equation 1)
2. $$ 2x - 3y = 3 $$  (Equation 2)

We can solve this system using substitution or elimination. Let's use elimination. First, we can multiply Equation 2 by 3 to align the coefficients of $$ y $$:
$$
6x - 9y = 9 \quad \text{(Equation 3)}
$$

Now we can add Equation 1 and Equation 3:
$$
(5x + 3y) + (6x - 9y) = 4 + 9
$$
This simplifies to:
$$
11x - 6y = 13
$$

Now we can solve for $$ y $$ in terms of $$ x $$ using Equation 1:
$$
3y = 4 - 5x \implies y = \frac{4 - 5x}{3}
$$

Substituting this expression for $$ y $$ into Equation 2:
$$
2x - 3\left(\frac{4 - 5x}{3}\right) = 3
$$
This simplifies to:
$$
2x - (4 - 5x) = 3
$$
$$
2x - 4 + 5x = 3
$$
$$
7x - 4 = 3
$$
$$
7x = 7 \implies x = 1
$$

Now substituting $$ x = 1 $$ back into the expression for $$ y $$:
$$
y = \frac{4 - 5(1)}{3} = \frac{4 - 5}{3} = \frac{-1}{3}
$$

Thus, the values are:
$$
x = 1, \quad y = -\frac{1}{3}
$$

### Summary of Results
- Part a: $$ p = 5, \quad q = 3 $$
- Part b: $$ x = \frac{5}{2}, \quad y = -\frac{5}{3} $$
- Part c: $$ x = 1, \quad y = -\frac{1}{3} $$
p = 5, q = 3 x = 5/2, y = -5/3 x = 1, y = -1/3

a. Find and if
b. Find and if
c. Find the real numbers and if

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a. Find and if b. Find and if c. Find the real numbers and if