The given equation is written in slope-intercept form, and the slope of the line is $\displaystyle -1/2$. The slope of a perpendicular line is the negative reciprocal of the given line. The negative reciprocal here is $\displaystyle 2$. Therefore, the correct equation is:

$\displaystyle \small y=2x+4$

The equation $\displaystyle ax + by = c$ can be rewritten as follows:

$\displaystyle ax + by = c$

$\displaystyle by = -ax + c$

$\displaystyle y = -\frac{a}{b}x + \frac{c}{b}$

This is the slope-intercept form, and the line has slope $\displaystyle -\frac{a}{b}$. 

The line of the equation $\displaystyle \frac{1}{2}x + \frac{1}{5}y =21$ therefore has slope

$\displaystyle m = -\frac{\frac{1}{2}}{\frac{1}{5}} =- \frac{1}{2} \div \frac{1}{5}= - \frac{1}{2} \cdot \frac{5}{1}= - \frac{5}{2}$

Since a line perpendicular to this one must have a slope that is the opposite reciprocal of $\displaystyle - \frac{5}{2}$, we are looking for a line with slope $\displaystyle \frac{2}{5}$.

The slopes of the lines in the four choices are as follows:

$\displaystyle 5x-2y =- 77$; $\displaystyle m = -\frac{5}{-2} = \frac{5}{2}$

$\displaystyle 5x+2y =- 77$; $\displaystyle m = -\frac{5}{2}$

$\displaystyle 2x+5y =- 77$: $\displaystyle m = -\frac{2}{5}$

$\displaystyle 2x-5y =- 77$: $\displaystyle m = -\frac{2}{-5} = \frac{2}{5}$ - this is the correct one.

$\displaystyle 4x + 13 = 0$ can be rewritten as follows:

$\displaystyle 4x = -13$

$\displaystyle x= -\frac{13}{4}$

Any line with equation $\displaystyle x = a$ is vertical and has undefined slope; a line perpendicular to this is horizontal and has slope 0, and can be written as $\displaystyle y = b$. The only choice that does not have an $\displaystyle x$ is $\displaystyle 4y + 17 = 0$, which can be rewritten as follows:

$\displaystyle 4y + 17 = 0$

$\displaystyle 4y = -17$

$\displaystyle y = - \frac{17}{4}$

This is the correct choice.

The equation $\displaystyle ax + by = c$ can be rewritten as follows:

$\displaystyle ax + by = c$

$\displaystyle by = -ax + c$

$\displaystyle y = -\frac{a}{b}x + \frac{c}{b}$

This is the slope-intercept form, and the line has slope $\displaystyle -\frac{a}{b}$. 

The line of the equation $\displaystyle 0.6x + 0.4y = 177$ has slope

$\displaystyle m = -\frac{0.6}{0.4} =- \frac{3}{2}$

Since a line perpendicular to this one must have a slope that is the opposite reciprocal of $\displaystyle -\frac{3}{2}$ , we are looking for a line that has slope $\displaystyle \frac{2}{3}$.

The slopes of the lines in the four choices are as follows:

$\displaystyle 3x+ 2y = 68$: $\displaystyle m = -\frac{3}{2}$

$\displaystyle 3x-2y = 68$: $\displaystyle m = -\frac{3}{-2} = \frac{3}{2}$

$\displaystyle 2x+ 3y = 68$: $\displaystyle m = -\frac{2}{3}$

$\displaystyle 2x- 3y = 68$: $\displaystyle m = -\frac{2}{-3} = \frac{2}{3}$ - the correct choice.

First, we need to find the slope of the above line. 

The slope of a line. given two points $\displaystyle (x_{1}, y_{1}), (x_{2}, y_{2})$ can be calculated using the slope formula:

$\displaystyle m = \frac{y_{2}-y_{1}}{x_{2}-x _{1}}$

Set $\displaystyle x_{1}=-6, y_{1}=x_{2}= 0, y_{2}=18$:

$\displaystyle m = \frac{18-0}{0-(-6)} = \frac{18}{6} = 3$

The slope of a line perpendicular to it has as its slope the opposite of the reciprocal of 3, which would be $\displaystyle m = -\frac{1}{3}$. Since we want this line to have the same $\displaystyle y$-intercept as the first line, which is the point $\displaystyle (0,18)$, we can substitute $\displaystyle m = -\frac{1}{3}$ and $\displaystyle b = 18$ into the slope-intercept form of the equation:

$\displaystyle y = mx + b$

$\displaystyle y = - \frac{1}{3}x + 18$

We calculate the slopes of the lines using the slope formula.

The slope of line $\displaystyle l$ is 

$\displaystyle m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{-3 - (-6)}{4-8} = \frac{3}{-4} = - \frac{3}{4}$.

The slope of line $\displaystyle k$ is 

$\displaystyle m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{-2 - 4}{5- (-3)} = \frac{-6}{8} = - \frac{3}{4}$.

The lines have the same slope, so either they are distinct parallel lines or one and the same line. One way to check for the latter situation is to find the slope of the line connecting one point on $\displaystyle l$ to one point on $\displaystyle k$ - if the slope is also $\displaystyle - \frac{3}{4}$, the lines coincide. We will use $\displaystyle (8, -6)$ and $\displaystyle (5, -2)$:

$\displaystyle m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{-2 - (-6)}{5-8} = \frac{4}{-3} = - \frac{4}{3}$.

The lines are therefore distinct and parallel.

We calculate the slopes of the lines using the slope formula.

The slope of line $\displaystyle l$ is 

$\displaystyle m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{-3 - (-6)}{4-8} = \frac{3}{-4} = - \frac{3}{4}$.

The slope of line $\displaystyle k$ is 

$\displaystyle m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{12 - 9}{-16- (-12)} = \frac{3}{-4} = - \frac{3}{4}$.

The lines have the same slope, so either they are distinct, parallel lines or one and the same line. One way to determine which is the case is to find the equations.

Line $\displaystyle l$, the line through $\displaystyle (4, -3)$ and $\displaystyle (8, -6)$, has equation

$\displaystyle y - (-3)= -\frac{3}{4} (x - 4)$

$\displaystyle y+3= -\frac{3}{4} x +3$

$\displaystyle y = -\frac{3}{4} x$

Line $\displaystyle k$, the line through $\displaystyle (-12, 9)$ and $\displaystyle (-16, 12)$, has equation

$\displaystyle y - 9= -\frac{3}{4} [x - (-12)]$

$\displaystyle y - 9= -\frac{3}{4} (x +12)$

$\displaystyle y - 9= -\frac{3}{4} x - 9$

$\displaystyle y = -\frac{3}{4} x$

The lines have the same equation, making them one and the same.

First, we need to find the slope of the above line. 

The slope of a line. given two points $\displaystyle (x_{1}, y_{1}), (x_{2}, y_{2})$ can be calculated using the slope formula:

$\displaystyle m = \frac{y_{2}-y_{1}}{x_{2}-x _{1}}$

Set $\displaystyle x_{1}=-4, y_{1}=x_{2}= 0, y_{2}=8$:

$\displaystyle m = \frac{8-0}{0-(-4)} = \frac{8}{4} = 2$

A line parallel to this line also has slope $\displaystyle m = 2$. Since it passes through the origin, its $\displaystyle y$-intercept is $\displaystyle (0,0)$, and we can substitute $\displaystyle m = 2,b=0$ into the slope-intercept form of the equation:

$\displaystyle y = mx+b$

$\displaystyle y = 2x + 0$

$\displaystyle y = 2x$

We find the slope of each line by putting each equation in slope-intercept form, $\displaystyle y =mx + b$, and examining the coefficient of $\displaystyle x$.

$\displaystyle y = 4x + 9$ is already in slope-intercept form; its slope is $\displaystyle m = 4$.

To get $\displaystyle 8x + 2y = 13$ in slope-intercept form we solve for $\displaystyle y$:

$\displaystyle 8x + 2y = 13$

$\displaystyle 8x + 2y -8x = 13 -8x$

$\displaystyle 2y =-8x+ 13$

$\displaystyle \frac{2y }{2}=\frac{-8x+ 13}{2}$

$\displaystyle y=\frac{-8x }{2} + \frac{ 13}{2}$

$\displaystyle y=-4x + \frac{ 13}{2}$

The slope of this line is $\displaystyle m =-4$.

The slopes are not equal so we can eliminate both "parallel" and "identical" as choices.

Multiply the slopes together:

$\displaystyle -4 \times 4 = -16$

The product of the slopes of the lines is not $\displaystyle -1$, so we can eliminate "perpendicular" as a choice.

The correct response is "neither".

We find the slope of each line by putting each equation in slope-intercept form $\displaystyle y =mx + b$ and examining the coefficient of $\displaystyle x$.

$\displaystyle y = 3x + 7$ is already in slope-intercept form; its slope is $\displaystyle m = 3$.

To get $\displaystyle 3x + 9y = 10$ into slope-intercept form we solve for $\displaystyle y$:

$\displaystyle 3x + 9y = 10$

$\displaystyle 3x + 9y - 3x = 10 - 3x$

$\displaystyle 9y = - 3x + 10$

$\displaystyle \frac{9y }{9} =\frac{ - 3x + 10 }{9}$

$\displaystyle y = \frac{-3x}{9}+ \frac{10}{9}$

$\displaystyle y =- \frac{1 }{3}x+ \frac{10}{9}$

The slope of this line is $\displaystyle m = - \frac{1}{3}$.

The slopes are not equal so we can eliminate both "parallel" and "one and the same" as choices.

Multiply the two slopes together:

$\displaystyle 3 \times \left ( - \frac{1}{3} \right ) = -1$

The product of the slopes of the lines is $\displaystyle -1$, making the lines perpendicular.