The complex conjugate of a complex number $\displaystyle a + bi$ is $\displaystyle a - bi$, so $\displaystyle 4+7i$ has $\displaystyle 4-7i$ as its complex conjugate; the sum of the two numbers is

$\displaystyle (4+7i)+ (4-7i) = 4+4 + 7i - 7i = 8$

The complex conjugate of a complex number $\displaystyle a + bi$ is $\displaystyle a - bi$, so the complex conjugate of $\displaystyle 3 + 7i$ is $\displaystyle 3 - 7i$. Multiply this by $\displaystyle 4i$:

$\displaystyle (3-7i) \cdot 4i = 3 \cdot 4i - 7 i \cdot 4i$

$\displaystyle = 12i - 7 \cdot 4 \cdot i \cdot i$

$\displaystyle = 12i -28 \cdot (-1)$

$\displaystyle = 12i + 28$

$\displaystyle = 28 + 12i$

The complex conjugate of a complex number $\displaystyle a + bi$ is $\displaystyle a - bi$. Since $\displaystyle 8=8+0i$, its complex conjugate is $\displaystyle 8-0i = 8$ itself. Multiply this by $\displaystyle 5i$:

$\displaystyle 8 \cdot 5i = 40 i$

The complex conjugate of a complex number $\displaystyle a + bi$ is $\displaystyle a - bi$. Since $\displaystyle 7i = 0+7i$, its complex conjugate is $\displaystyle 0-7i = -7i$.

Multiply this by $\displaystyle 3i$:

Recall that by definition $\displaystyle i^2=-1$.

$\displaystyle -7i \cdot 3i = -7 \cdot 3 \cdot i \cdot i = -21 (-1) = 21$

FOIL the product out:

$\displaystyle \left (5 + i \right )\left (5 + 2i \right )$

To FOIL multiply the first terms from each binomial together, multiply the outer terms of both terms together, multiply the inner terms from both binomials together, and finally multiply the last terms from each binomial together.

$\displaystyle =5 \cdot 5 + 5 \cdot 2i + 5 \cdot i + i\cdot 2i$

$\displaystyle =25 + 10i + 5 i + 2i^{2}$

Recall that i is an imaginary number and by definition $\displaystyle i^2=-1$. Substituting this into the function is as follows.

$\displaystyle =25 + 10i + 5 i + 2 (-1)$

$\displaystyle =25 -2 + 10i + 5 i$

$\displaystyle =23 + 15i$

The vertical asymptote of an inverse variation function is the vertical line of the equation $\displaystyle x = a$, where $\displaystyle a$ is the value for which the expression is not defined. To find $\displaystyle a$, set the denominator to $\displaystyle 0$ and solve for $\displaystyle x$:

$\displaystyle 9x+4 = 0$

$\displaystyle 9x= -4$

$\displaystyle x= -\frac{4}{9}$ is the equation of the asymptote.

The $\displaystyle x$-intercept of the graph of an equation is the point at which it intersects the $\displaystyle y$-axis. Its $\displaystyle y$-coordinate is 0, so set $\displaystyle y = 0$ and solve for $\displaystyle x$:

$\displaystyle y = \frac{5}{2x-7}$

$\displaystyle 0= \frac{5}{2x-7}$

$\displaystyle 0\cdot (2x-7) = \frac{5}{2x-7} \cdot (2x-7)$

$\displaystyle 0 = 5$

This is identically false, so the graph has no $\displaystyle x$-intercept.

The $\displaystyle y$-intercept of the graph of an equation is the point at which it intersects the $\displaystyle x$-axis. Its $\displaystyle x$-coordinate is 0, so set $\displaystyle x = 0$ and solve for $\displaystyle y$:

$\displaystyle y = \frac{3}{5x-8}$

$\displaystyle y = \frac{3}{5 (0)-8}$

$\displaystyle y = \frac{3}{0-8}$

$\displaystyle y =- \frac{3}{8}$

$\displaystyle \left ( 0, -\frac{3}{8}\right )$ is the $\displaystyle y$-intercept.

The graph of $\displaystyle y = \frac{6}{8x-2}$ does not have an $\displaystyle x$-intercept. If it did, then it would be the point on the graph with $\displaystyle y$-coordinate 0. If we were to make this substitution, the equation would be

$\displaystyle 0= \frac{6}{8x-2}$

and 

$\displaystyle 0 \cdot (8x-2)= \frac{6}{8x-2} \cdot (8x-2)$

$\displaystyle 0=6$

This is identically false, so the graph has no $\displaystyle x$-intercept. Therefore, the line cannot exist as described.

Substitute $\displaystyle \frac{4}{x}$ for $\displaystyle y$ in the second equation:

$\displaystyle x+3\cdot \frac{4}{x} = 4$

$\displaystyle x+ \frac{12}{x} = 4$

$\displaystyle x+ \frac{12}{x}- 4 = 4 - 4$

$\displaystyle x- 4 + \frac{12}{x}= 0$

$\displaystyle \left (x- 4 + \frac{12}{x} \right )\cdot x = 0 \cdot x$

$\displaystyle x^{2}- 4 x + 12 = 0$

The discriminant of this quadratic expression is $\displaystyle b^{2} - 4ac$, where $\displaystyle a=1, b= -4, c=12$; this is

$\displaystyle b^{2} - 4ac = (-4)^{2}-4 \cdot 1 \cdot 12 = 16 -48 = -32$.

The discriminant being negative, there are no real solutions to this quadratic equation. Consequently, there are no points of intersection of the graphs of the two equations on the coordinate plane.