what is the slope of any line perpendicular to the line 4/(3x+2y)=-14?

What you want to do first is find the equation of the line. We'll put it into the form y=mx+b, where m is the slope and b is the y-intercept.

$$\frac{4}{3x+2y} = -14$$

Well, we want y all by itself, so what should we do? Let's multiply both sides of the equation by the denominator!

$$\frac{4}{3x+2y} * (3x+2y) = -14 * (3x+2y) \Rightarrow 4 = -42x - 28y$$

I went ahead and used the distributive property to help simplify the problem. Now let's keep making y alone! So, y doesn't like x, so we'll take him away by adding 42x to each side of the equation.

$$4 + 42x = -42x - 28x +42x \Rightarrow 4 + 42x = -28y$$

The y still isn't alone. He has that -28 right next to him! Let's divide both sides by -28 to get rid of him.

$$\frac{4+42x}{-28} = \frac{-28y}{-28} \Rightarrow -\frac{1}{7} - \frac{3}{2}x = y$$

Now y is alone! let's rewrite this as the whole y=mx+b form.

$$y = -\frac{3}{2}x-\frac{1}{7}$$

Remember, m is the slope. So, m=-3/2!

Now, to do the perpendicular of a line, just flip the slope and change the sign! So, the new perpendicular slope equals 2/3!

First rewrite this as follows:

Multiply through by 3x + 2y

4 = -14(3x + 2y)

4 = -42x - 28y

Ad 28y to both sides and subtract 4 from both sides

28y = -42x - 4

Divide through by 28

y = -(42/28)x - 4/28  or y = -(3/2)x - 1/7

The slope of this line is -(3/2) so the slope of a line perpendicular to this is +(2/3)

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