3. Given that $$ i^{2}=-1 $$, and $$ \frac{2 i-1}{3 i+5}+\frac{2 i-3}{i-1}=\frac{a(i-b)}{i+c} $$, what is the value of $$ a+b+c $$ ? A. 75 B. 16 C. 14 D. 6 E. -16

Question

UpStudy AI · Accepted Answer

We begin with the given equation

$$
\frac{2i-1}{3i+5} + \frac{2i-3}{i-1} = \frac{a(i-b)}{i+c}.
$$

**Step 1. Combine the left-hand side fractions**

Find a common denominator for

$$
\frac{2i-1}{3i+5}+\frac{2i-3}{i-1}.
$$

The common denominator is $$(3i+5)(i-1)$$. Thus the combined expression is

$$
\frac{(2i-1)(i-1) + (2i-3)(3i+5)}{(3i+5)(i-1)}.
$$

**Step 2. Expand the Numerators**

First, expand $$(2i-1)(i-1)$$:

$$
(2i-1)(i-1)=2i(i-1)-1(i-1)=2i^2-2i-i+1=2i^2-3i+1.
$$

Next, expand $$(2i-3)(3i+5)$$:

$$
(2i-3)(3i+5)=2i(3i+5)-3(3i+5)=6i^2+10i-9i-15=6i^2+i-15.
$$

Then, add the two results:

$$
2i^2-3i+1+6i^2+i-15 = (2i^2+6i^2)+(-3i+i)+(1-15)=8i^2-2i-14.
$$

Factor out 2:

$$
8i^2-2i-14 = 2\bigl(4i^2 - i - 7\bigr).
$$

**Step 3. Simplify the Denominator**

Multiply out the denominator $$(3i+5)(i-1)$$:

$$
(3i+5)(i-1)=3i(i-1)+5(i-1)=3i^2-3i+5i-5=3i^2+2i-5.
$$

Since we know $$ i^2=-1 $$, substitute:

$$
3(-1)+2i-5=-3+2i-5=2i-8=2(i-4).
$$

**Step 4. Write the Simplified Expression**

The fraction becomes:

$$
\frac{2\bigl(4i^2-i-7\bigr)}{2(i-4)}=\frac{4i^2-i-7}{i-4}.
$$

Now use $$ i^2=-1 $$ to simplify the numerator:

$$
4(-1)-i-7=-4-i-7=-i-11.
$$

Thus, the left-hand side is

$$
\frac{-i-11}{i-4}.
$$

**Step 5. Equate to the Right-Hand Side**

We have

$$
\frac{-i-11}{i-4}=\frac{a(i-b)}{i+c}.
$$

For the expressions to be equal for all valid $$ i $$ (where $$ i $$ is the imaginary unit), the denominators must match. This suggests

$$
i+c=i-4 \quad \Longrightarrow \quad c = -4.
$$

Then, equate the numerators:

$$
a(i-b)=-i-11.
$$

Expand the left-hand side:

$$
ai-ab = -i-11.
$$

Now, equate the coefficients of like terms:

- Coefficient of $$ i $$: 
  $$
  a=-1.
  $$
- Constant term:
  $$
  -ab=-11 \quad \Longrightarrow \quad -(-1)b=-11 \quad \Longrightarrow \quad b=-11.
  $$

**Step 6. Compute $$a+b+c$$**

Now that we have found $$a=-1$$, $$b=-11$$, $$c=-4$$, we compute:

$$
a+b+c=(-1)+(-11)+(-4)=-16.
$$

Thus, the value of $$ a+b+c $$ is

$$
\boxed{-16}.
$$
$$ a + b + c = -16 $$

Given that , and , what is the value of ?
A. 75
B. 16
C. 14
D. 6
E. -16

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Given that , and , what is the value of ? A. 75 B. 16 C. 14 D. 6 E. -16