Question

What is the equation, in slope-intercept form, of the line that is perpendicular to the line

y – 4 = –(x – 6) and passes through the point (−2, −2)?

y = –x –
y = –x +
y = x – 1
y = x + 1

Answer 1

`\bf \textit{perpendicular, negative-reciprocal slope for slope}\quad \cfrac{a}{b}\\\\ slope=\cfrac{a}{{{ b}}}\qquad negative\implies -\cfrac{a}{{{ b}}}\qquad reciprocal\implies - \cfrac{{{ b}}}{a}\\\\ -------------------------------\\\\ \textit{so if this equation has a slope of -1, a perpendicular line will have} \\\\\\ \cfrac{-1}{1}\qquad negative\implies \cfrac{1}{1}\qquad reciprocal\implies \cfrac{1}{1}\implies 1`

so.. what's the equation of a line whose slope is 1 and passes through -2,-2?

Answer 2

the equation of the line is equal to

`y=x`

Explanation:

Step 1

Find the slope of the line that is perpendicular to the given line

we know that

If two lines are perpendicular

then

the product of their slopes is equal to minus one

so

`m1*m2=-1`

we have the given line

`y-4=-(x-6)`

this is the equation of the line into point-slope form

the slope is equal to

`m1=-1`

`m1*m2=-1`

Find m2

`m2=-1/m1`

`m2=-1/(-1)=1`

The slope of the line perpendicular to the given line is equal to
`1`

Step 2

Find the equation of the line in slope-intercept form

we know that

the equation of the line in slope-intercept form is equal to

`y=mx+b`

where

m is the slope

b is the y-intercept

in this problem we have

`m=1`

point
`(-2,-2)`

substitute and find the value of b

`-2=(1)(-2)+b`

`-2=-2+b`

`b=0`

the equation of the line is equal to

`y=x`