The slope of m is equal to   ^y2-y1/_x2-x1  =  ^2-4/_5-1 = ^-1/₂                                  

Since line p is perpendicular to line m, this means that the products of the slopes of p and m must be –1:

(slope of p) * (^-1/₂) = -1

Slope of p = 2

So we must choose the equation that has a slope of 2. If we rewrite the equations in point-slope form (y = mx + b), we see that the equation 2x – y = 3 could be written as y = 2x – 3. This means that the slope of the line 2x – y =3 would be 2, so it could be the equation of line p. The answer is 2x – y = 3.

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, $\displaystyle -\frac{2}{3}$.

Parallel lines have slopes that are opposite reciprocals. Any line perpendicular to $\displaystyle y=3x+9$ must have a slope of $\displaystyle -\frac{1}{3}$.  The only line that doesn't have this slope is $\displaystyle y=\frac{1}{3}+2$, which has a slope of $\displaystyle \frac{1}{3}$.

The given key word is vertical.  If a perpendicular line that intersects the vertical line, the slope must be zero, and the line would be a horizontal line.  The horizontal line will intersect the point at $\displaystyle (2,1)$, which means that the y-value will never change.

The answer is:  $\displaystyle y=1$

First, we want to identify the slope of our given line. In slope-intercept form, a line's slope $\displaystyle m$ is the coefficient of $\displaystyle x$. 

$\displaystyle y=mx+b$

The slope of our given line is $\displaystyle \frac{4}{3}$. 

Now, we can start to find the perpendicular line. Remember that in order for two lines to be perpendicular, their slopes must be negative reciprocals of each other. 

In this case, we want to find a line whose slope is the negative reciprocal of $\displaystyle \frac{4}{3}$. So, we want a line whose slope is $\displaystyle -\frac{3}{4}$. 

With this in mind, the only answer choice with a slope of $\displaystyle -\frac{3}{4}$ is $\displaystyle y = -\frac{3}{4}x + 2$. 

Therefore, the perpendicular line is $\displaystyle y = -\frac{3}{4}x + 2$. 

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:

$\displaystyle y = 2x + 10$

First, put the equation of the line given into slope-intercept form by solving for y. You get y = -2x +5, so the slope is –2. Perpendicular lines have opposite-reciprocal slopes, so the slope of the line we want to find is 1/2. Plugging in the point given into the equation y = 1/2x + b and solving for b, we get b = 6. Thus, the equation of the line is y = ½x + 6. Rearranged, it is –x/2 + y = 6.

The slope of line m is the (y₂ - y₁) / (x₂ - x₁) = (5-7) / (-2 - 3)

= -2 / -5 

= 2/5

To find the slope of a line perpendicular to a given line, we must take the negative reciprocal of the slope of the given line.

Thus the slope of line k is the negative reciprocal of 2/5 (slope of line m), which is -5/2.