Question

Write an equation of the line containing the given point and perpendicular to the given line:

​(4​, - 7​); 9x+5y=3

Answer 1

`y=(5)/(9)x+(-83)/(9)`

Explanation:

Given equation of line:

`9x+5y=3`

To find the equation of line perpendicular to the line of the given equation and passes through point (4,-7).

Writing the given equation of line in standard form.

Subtracting both sides by
`9x`

`9x-9x+5y=3-9x`

`5y=3-9x`

Dividing both sides by 5.

`(5y)/(5)=(3)/(5)-(9x)/(5)`

`y=(3)/(5)-(9x)/(5)`

Rearranging the equation in standard form
`y=mx+b`

`y=-(9x)/(5)+(3)/(5)`

Applying slope relationship between perpendicular lines.

`m_1=-(1)/(m_2)`

where
`m_1` and
`m_2` are slopes of perpendicular lines.

For the given equation in the form
`y=mx+b` the slope
`m_2`can be found by comparing
`y=-(9x)/(5)+(3)/(5)` with standard form.

∴
`m_2=-(9)/(5)`

Thus slope of line perpendicular to this line
`m_1` would be given as:

`m_1=-(1)/(-(9)/(5))`

∴
`m_1=(5)/(9)`

The line passes through point (4,-7)

Using point slope form:

`y-y_1=m(x-x_1)`

Where
`(x_1,y_1)\rightarrow (4,-7)` and
`m=m_1=(5)/(9)`

So,

`y-(-7)=(5)/(9)(x-4)`

Using distribution.

`y+7=((5)/(9)x)-((5)/(9)* 4)`

`y+7=(5)/(9)x-(20)/(9)`

Subtracting 7 to both sides.

`y+7-7=(5)/(9)x-(20)/(9)-7`

Taking LCD to subtract fractions

`y=(5)/(9)x-(20)/(9)-(63)/(9)`

`y=(5)/(9)x+((-20-63))/(9)`

`y=(5)/(9)x+(-83)/(9)`

`y=(5)/(9)x-(83)/(9)`

Thus, the equation of line in standard form is given by:

`y=(5)/(9)x-(83)/(9)`