Question

How do you write an equation (a) in slope-intercept form and (b) in standard form for the line passing through (3,7) and perpendicular to 4x+5y=1?

Answer 1

Slope-intercept : `y = 5/4 x + 13/4`
Standard : `-5x + 4y = 13`

Explanation:

To write a linear equation in any form, you need at least two of the following three pieces of information: a point, a different point, and/or a slope. We are already given a single point, `(3, 7)`.

Note that if we have a line `y = mx + b` in slope-intercept form, its slope is given by `m`. The line perpendicular to this line will have a slope of `-1/m`. So let's put our provided equation in slope-intercept form.

`4x + 5y = 1`
`5y = 1 - 4x`
`y = 1/5 - 4/5 x = -4/5 x + 1/5`

The slope of the given line is then `-4/5`. The slope of the line perpendicular to this must be `5/4`. Now we have both a slope and a point.

To find our equation in slope-intercept form, we take the base form `y = mx + b` and plug in our slope of `m = 5/4` and our point of `(x, y) = (3, 7)` to find `b`.

`y = mx + b`
`y = 5/4 x + b`
`7 = 5/4 (3) + b`
`7 = 15/4 + b`
`7 - 15/4 = b`
`28/4 - 15/4 = b`
`13/4 = b`.

Having found `b`, we now know our equation in slope-intercept form. That is, `y = 5/4 x + 13/4`.

The standard form of a linear equation is `ax + by = c`, where `a, b, c` are constants. It is easy to move from slope-intercept form into standard form.

`y = 5/4 x + 13/4`
`4y = 5x + 13`
`-5x + 4y = 13`

Thus, our answer in standard form is `-5x + 4y = 13`.