Question

What is an equation in slope-intercept form of the line that is perpendicular to the graph of `y= -4x+5` and passes through (1, 1)?

Answer 1

The equation of line in slope-intercept form is `y=1/4 x +3/4`

Explanation:

Slope of the line, `y= -4 x +5 or ; [y=mx+c]`

is `m_1= -4` [Compared with slope-intercept form of equation]

The product of slopes of the pependicular lines is `m_1*m_2=-1`

`:.m_2=(-1)/-4=1/4`. The equation of line passing through

`(x_1=1,y_1=1)` having slope of `m_2` is `y-y_1=m_2(x-x_1)`.

`:. y-1=1/4(x-1) or y = 1/4 x -1/4+1 or y= 1/4 x +3/4`.

Equation of line in slope-intercept form is `y=1/4 x +3/4` [Ans]

Answer 2

`y=1/4x+34`

Explanation:

The given equation `y=color(green)(-4)x+color(red)5`
is in slope-vertex form with
slope `color(green)m=color(green)(-4)`, and
y-intercept `color(red)b=color(red)5`

If a line has a slop of `color(green)m`
then every line perpendicular to it has a slope of `color(blue)(-1/m)`.

Therefore all lines perpendicular to `y=color(green)(-4)+5`
will have a slope of `color(blue)(1/4)`
and will have an equation in slope-intercept form:
`color(white)("XXX")y=color(blue)(1/4)x+color(magenta)c` for some constant (the y-intercept) `color(magenta)c`

If `(color(brown)x,color(lime)y)=(color(brown)1,color(lime)1)` is a solution for the required line with this form,
then
`color(white)("XXX")color(lime)1=color(blue)(1/4) * color(brown)1 + color(magenta)c`

`color(white)("XXX")rArr color(magenta)c=3/4`

Therefore the resolved equation for the given line is
`color(white)("XXX")y=1/4x+3/4`