Question

What is the equation of the line perpendicular to `y = 3x -2` and passes through point (3,4)?

Answer 1

`y=-1/3x+5`

Explanation:

`"given a line with slope m then the slope of a line "`
`"perpendicular to it is"`

`•color(white)(x)m_(color(red)"perpendicular")=-1/m`

`y=3x-2" is in "color(blue)"slope-intercept form"`

`•color(white)(x)y=mx+b`

`"where m is the slope and b the y-intercept"`

`y=3x-2rArrm=3`

`rArrm_(color(red)"perpendicular")=-1/3`

`rArry=-1/3x+blarr" is the partial equation"`

`"to find b substitute "(3,4)" into the partial equation"`

`4=-1+brArrb=5`

`rArry=-1/3x+5larrcolor(red)" in slope-intercept form"`

Answer 2

`y=color(red)(-1/3)x+color(blue)(5)`

Explanation:

Let us take the equation of the line as:
`y=color(red)(m)x+color(blue)(b)` , where m is the slope of the line.

Since the 2 lines are perpendicular, we use the negative reciprocal rule where:

• The product of the lines' slopes should be equal to -1.

From this rule, we get that:
`m*3=-1`

We divide both sides by 3 to get the slope `m`:
`mcancel(*3)/cancel 3=-1/3`
`-> color(red)(m=-1/33`

Equation of line:
`y=-1/3x+b`

To find the y-intercept `b`, we will have to put the values:
Let us call the given point A:
`A(3,4)`

Since A lies on the line:
`y_A=-1/3*x_A+b`
`4=-1/cancel3cancel(*3)+b`
`4=-1+b`

Adding 1 to both sides to isolate y-intercept b:
`4+1=cancel(-1)cancel(+1)+b`
`-> color(blue)(b=5`

Putting the values of `color(red)(m` and `color(blue)(b`
`y=color(red)(m)x+color(blue)(b)`
`rArry=color(red)(-1/3)x+color(blue)(5)`