What is the equation of the line perpendicular to $y = 3 x - 7$ $y =3x- 7$ that contains (6, 8)?

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Answer 1 · 2016-12-29T18:09:55

$(y - 8) = - \frac{1}{3} (x - 6)$ $(y - 8) = -1/3(x - 6)$

or

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

Because the line given in the problem is in the slope intercept form we know the slope of this line is $3$ $color(red)(3)$

The slope-intercept form of a linear equation is:

$y = m x + b$ $y = color(red)(m)x + color(blue)(b)$

Where $m$ $color(red)(m)$ is the slope and $b$ $color(blue)(b$ is the y-intercept value.
This is a weighted average problem.

Two perpendicular lines have a negative inverse slope of each other.

The line perpendicular to a line with slope $m$ $color(red)(m)$ has a slope of $- \frac{1}{m}$ $color(red)(-1/m)$ .

Therefore, the line we are looking for has a slope of $- \frac{1}{3}$ $color(red)(-1/3)$ .

We can now use the point-slope formula to find the equation of the line we are looking for.

The point-slope formula states: $(y - y_{1}) = m (x - x_{1})$ $(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))$

Where $m$ $color(blue)(m)$ is the slope and $color(red)(((x_1, y_1)))$ is a point the line passes through.

We can substitute the slope we calculate and the point we were given to give the equation we are looking for:

$(y - color(red)(8)) = color(blue)(-1/3)(x - color(red)(6))$

If we want to put this in slope-intercept form we can solve for $y$ :

$y - color(red)(8) = color(blue)(-1/3)x - (color(blue)(-1/3) xx color(red)(6)))$

$y - color(red)(8) = color(blue)(-1/3)x - (-2)$

$y - color(red)(8) = color(blue)(-1/3)x + 2$

$y - color(red)(8) + 8 = color(blue)(-1/3)x + 2 + 8$

$y - 0 = color(blue)(-1/3)x + 10$

$y = -1/3x + 10$

What is the equation of the line perpendicular to y =3x- 7y=3x−7y =3x- 7 that contains (6, 8)?