Question

Write an equation in standard form for the line that passes through (5, -2) and is perpendicular to 3x-4y=12

Answer 1

The equation in standard form for the line that passes through (5, -2) and is perpendicular to 3x - 4y = 12 is 4x + 3y = 14

Solution:

Given that line that passes through (5, -2) and is perpendicular to 3x - 4y = 12

We have to find the equation of line

The slope intercept form is given as:

y = mx + c ------ eqn 1

Where "m" is the slope of line and "c" is the y-intercept

Let us first find the slope of line

Given equation of line is 3x - 4y = 12

On rearranging the above equation to slope intercept form,

3x - 4y = 12

4y = 3x - 12

`y = (3)/(4)x - 3`

On comparing the above equation with slope intercept form,

`m = (3)/(4)`

Thus the slope of line is
`m = (3)/(4)`

We know that product of slope of given line and slope of line perpendicular to given line is always -1

slope of given line x slope of line perpendicular to given line = -1

`\begin{array}{l}{(3)/(4) * \text { slope of line perpendicular to given line }=-1} \\\\ {\text {slope of line perpendicular to given line }=(-4)/(3)}\end{array}`

Now we have to find the equation of line with slope
`m = (-4)/(3)` and passes through (5, -2)

Substitute
`m = (-4)/(3)` and (x, y) = (5, -2) in eqn 1

`-2 = (-4)/(3)(5) + c\\\\-2 = (-20)/(3) + c\\\\-6 = -20 + 3c\\\\3c = 14\\\\c = (14)/(3)`

The required equation of line is:

Now substitute
`m = (-4)/(3)` and
`c = (14)/(3)`

`y = (-4)/(3)x + (14)/(3)`

The standard form of an equation is Ax + By = C

x and y are variables and A, B, and C are integers

Rewriting the above equation,

`y = (-4)/(3)x + (14)/(3)`

3y = -4x + 14

4x + 3y = 14

Thus the equation of line in standard form is found out