Perpendicular slopes are opposite reciprocals.

The given slope is found by converting the equation to the slope-intercept form.

$\displaystyle 4x - 3y = 6$

$\displaystyle y = \frac{4}{3}x - 2$ 

The slope of the given line is $\displaystyle m = \frac{4}{3}$ and the perpendicular slope is  $\displaystyle m = \frac{-3}{4}$.

We can use the given point and the new slope to find the perpendicular equation. Plug in the slope and the given coordinates to solve for the y-intercept.

$\displaystyle 6 = \frac{-3}{4}(4) + b$

$\displaystyle b = 9$

Using this y-intercept in slope-intercept form, we get out final equation: $\displaystyle y = \frac{-3}{4}x + 9$.

The definition of a perpendicular line is one that has a negative, reciprocal slope to another.

For this particular problem, we must first manipulate our initial equation into a more easily recognizable and useful form: slope-intercept form or $\displaystyle y=mx+b$.

$\displaystyle 5x+6y=18$

$\displaystyle 6y=-5x+18$

$\displaystyle y=-\frac{5}{6}x+6$

According to our $\displaystyle y=mx+b$ formula, our slope for the original line is $\displaystyle -\frac{5}{6}$. We are looking for an answer that has a perpendicular slope, or an opposite reciprocal. The opposite reciprocal of $\displaystyle -\frac{5}{6}$ is $\displaystyle \frac{6}{5}$. Flip the original and multiply it by $\displaystyle -1$. 

Our answer will have a slope of $\displaystyle \frac{6}{5}$. Search the answer choices for $\displaystyle \frac{6}{5}$ in the $\displaystyle m$ position of the $\displaystyle y=mx+b$ equation.

$\displaystyle \dpi{100} y = \frac{6}{5}x + 3$ is our answer. 

(As an aside, the negative reciprocal of 4 is $\displaystyle -\frac{1}{4}$. Place the whole number over one and then flip/negate. This does not apply to the above problem, but should be understood to tackle certain permutations of this problem type where the original slope is an integer.)

Putting the first equation in slope-intercept form yields $\displaystyle y=\frac{3}{2}x+\frac{3}{2}$.

A perpendicular line has a slope that is the negative inverse. In this case, $\displaystyle -\frac{2}{3}$.

In order to solve this problem, we need first to transform the equation from standard form to slope-intercept form:

$\displaystyle y = mx + b$

Transform the original equation to find its slope.

$\displaystyle 3x + 6y = 12$ 

First, subtract $\displaystyle 3x$ from both sides of the equation.

$\displaystyle 3x-3x + 6y = 12-3x$

Simplify and rearrange.

$\displaystyle 6y = -3x + 12$ 

Next, divide both sides of the equation by 6.

$\displaystyle \frac{6y}{6} = -\frac{3x}{6} + \frac{12}{6}$

$\displaystyle y = -\frac{1}{2}x+ 2$

The slope of our first line is equal to $\displaystyle -\frac{1}{2}$. Perpendicular lines have slopes that are opposite reciprocals of each other; therefore, if the slope of one is x, then the slope of the other is equal to the following: 

$\displaystyle -\frac{1}{x}$

Let's calculate the opposite reciprocal of our slope:

$\displaystyle -\frac{1}{2}\rightarrow -\left (-\frac{2}{1} \right )=2$

The slope of our line is equal to 2. We now have the following partial equation:

$\displaystyle y = 2x + b$

We are missing the y-intercept, $\displaystyle b$. Substitute the x- and y-values in the given point $\displaystyle (-2, 6)$ to solve for the missing y-intercept. 

$\displaystyle 6 = 2(-2) + b$

$\displaystyle 6 = -4 + b$ 

Add 4 to both sides of the equation.

$\displaystyle 6+4 = -4+4 + b$

$\displaystyle 10 = b$

Substitute this value into our partial equation to construct the equation of our line:

$\displaystyle y = 2x + 10$

A line segment has endpoints (0,4) and (5,6). To find the midpoint, use the midpoint formula:

X: (x₁+x₂)/2 = (0+5)/2 = 2.5 

Y: (y₁+y₂)/2 = (4+6)/2 = 5

The coordinates of the midpoint are (2.5,5).

You can find the midpoint of each coordinate by averaging them.  In other words, add the two x coordinates together and divide by 2 and add the two y coordinates together and divide by 2.

x-midpoint = (-3 + 5)/2 = 2/2 = 1

y-midpoint = (7 + -9)/2 = -2/2 = -1

(1,-1)

The correct answer is (6, 6). The midpoint formula is ((x₁ + x₂)/2),((y₁ + y₂)/2) So 1 + 11 = 12, and 12/2 = 6 for both the x and y coordinates.

midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)

             = ((–1 + 3)/2, (2 – 6)/2)

             = (2/2, –4/2)

             = (1,–2)

To solve this problem you will need to use the midpoint formula:

$\displaystyle midpoint = (\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2} )$

Plug in the given values for the endpoints of the segment: $\displaystyle (-1,4)$ and $\displaystyle (3,16)$.

$\displaystyle midpoint = (\frac{-1+3}{2},\frac{4+16}{2} ) = (\frac{2}{2}, \frac{20}{2}) = (1, 10)$

The midpoint is the point halfway between the two endpoints, so sum up the coordinates and divide by 2:

$\displaystyle (\frac{1+5}{2},\frac{4+12}{2})=(\frac{6}{2},\frac{16}{2})=(3,8)$