Convert the equation to slope intercept form to get y = –1/3x + 2.  The old slope is –1/3 and the new slope is 3.  Perpendicular slopes must be opposite reciprocals of each other:  m_{1 *} m₂ = –1

With the new slope, use the slope intercept form and the point to calculate the intercept: y = mx + b or 5 = 3(1) + b, so b = 2

So y = 3x + 2

Convert the given equation to slope-intercept form.

\(\displaystyle 3x + 5y = 15\)

\(\displaystyle 5y=-3x+15\)

\(\displaystyle y=-\frac{3}{5}+3\)

The slope of this line is \(\displaystyle -\frac{3}{5}\). The slope of the line perpendicular to this one will have a slope equal to the negative reciprocal.

The perpendicular slope is \(\displaystyle \frac{5}{3}\).

Plug the new slope and the given point into the slope-intercept form to find the y-intercept.

\(\displaystyle 2 = \frac{5}{3}(3) + b\)

\(\displaystyle 2=5+b\)

\(\displaystyle b = -3\)

So the equation of the perpendicular line is \(\displaystyle y = \frac{5}{3}x - 3\).

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 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, \(\displaystyle -\frac{2}{3}\).

To start, begin by dividing everything by \(\displaystyle 9\), this will get your equation into the format \(\displaystyle y=mx+b\).  This gives you:

\(\displaystyle y=\frac{1}{3}x+\frac{22}{9}\)

Now, recall that the slope of a perpendicular line is the opposite and reciprocal slope to its mutually perpendicular line.  Thus, if our slope is \(\displaystyle \frac{1}{3}\), then the perpendicular line's slope must be \(\displaystyle -3\).  Thus, we need to look at our answers to determine which equation has a slope of \(\displaystyle -3\).  Among the options given, the only one that matches this is \(\displaystyle 3y + 9x = 54\).  If you solve this for \(\displaystyle y\), you will get:

\(\displaystyle y = -3x + \frac{18}{3}\)

For two lines to be perpendicular their slopes must have a product of \(\displaystyle -1\).
\(\displaystyle -3*\frac{1}{3} = -1\) and so we see the correct answer is given by \(\displaystyle y = \frac{1}{3}x + 2\)

Perpendicular lines have slopes whose product is \(\displaystyle -1\).

Looking at our equations we can see that it is in slope-intercept form where the m value represents the slope of the line,

\(\displaystyle y=mx+b\).

In our case we see that

\(\displaystyle y = -9x + 2\) therefore, \(\displaystyle m=-9\).

Since 

\(\displaystyle -9*\frac{1}{9} = -1\) we see the only possible answer is 

\(\displaystyle y = \frac{1}{9}x +6\).