Question

What is the equation of the line that is perpendicular to `2y=5x-4` and has a `y`-intercept of `b=-3`?

Answer 1

`2x` + `5y` = ╼ `15`

Explanation:

Lines that are perpendicular have slopes which are
the `"Negative inverse"` of each other.

1) First find the slope of the given line.
2) Change its sign to the opposite and invert the fraction
3) Use the given point for the y intercept `b`

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1) Find the slope of the given line

To find the slope, write the equation of the given line in slope-intercept form
`y = mx + b`
where the value at `m` is the slope.

`2y=5x−4`
Solve for `y` by dividing all the terms on both sides by 2

`y = (5)/(2)x - 2`

This result means that the slope of the given line is `(5)/(2)`, which is the value at `m`

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2) The slope of the perpendicular line
is the "`"negative inverse"`" of `(5)/(2)`

To find the slope of the perpendicular line, invert the fraction and change its sign

The slope `m` of the perpendicular line will be `-` `(2)/(5)`

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3) Use the given y intercept for `b`

The formula for the perpendicular line is

`y = mx + b`

where `m` was calculated to be `-(2)/(5)`
and where `b` is given as `-3`

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4) Write the equation

`y = mx + b`

`y = - (2)/(5)x - 3`

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5) In Standard Form the equation for the perpendicular line is

`ax + by = c`

Change to Standard Form
`y = - (2)/(5)x - 3`

1) Multiply all the terms on both sides by 5 to clear the fraction

`5y = - 2x - 15`

2) Add `2x` to both sides

`2x` + `5y` = ╼ `15`

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Answer:

The equation of the perpendicular line:

`2x` + `5y` = ╼ `15`

Answer 2

`y=-2/5x-3`

Explanation:

`"the equation of a line in "color(blue)"slope-intercept form"` is.

`•color(white)(x)y=mx+b`

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

`"rearrange "2y=5x-4" into this form"`

`rArry=5/2x-2larrcolor(blue)(m=5/2)`

`"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`

`rArrm_(color(red)"perpendicular")=-1/(5/2)=-2/5`

`"here "b=-3`

`rArry=-2/5x-3larrcolor(red)"in slope-intercept form"`