Question

Geometry
Solve both and show work

Answer 1

To find the equation of a line perpendicular to another line, we need to determine the slope of the original line and then take the negative reciprocal of that slope.

The given line is y + 3x = 13. We can rearrange it to the slope-intercept form (y = mx + b) by isolating y:

y = -3x + 13

The slope of this line is -3. The negative reciprocal of -3 is 1/3, which will be the slope of the perpendicular line.

Now, we have the slope (1/3) and a point (6, 10) that the perpendicular line passes through. We can use the point-slope form (y - y1 = m(x - x1)) to find the equation of the line.

Substituting the values into the point-slope form:

y - 10 = (1/3)(x - 6)

Distributing 1/3 to (x - 6):

y - 10 = (1/3)x - 2

Adding 10 to both sides to isolate y:

y = (1/3)x - 2 + 10

Simplifying:

y = (1/3)x + 8

Therefore, the equation of the line perpendicular to y + 3x = 13 and passing through the point (6, 10) is y = (1/3)x + 8.

Answer 2

y = - 5x - 1 and y =
`(1)/(3)` x + 8

Explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

14

y = - 5x + 4 ← is in slope- intercept form

with slope m = - 5

• Parallel lines have equal slopes , then

y = - 5x + c ← is the partial equation

to find c substitute P (- 1, 4 ) into the partial equation

4 = - 5(- 1) + c = 5 + c ( subtract 5 from both sides )

- 1 = c

y = - 5x - 1 ← equation of parallel line

15

y + 3x = 13 ( subtract 3x from both sides )

y = - 3x + 13 ← in slope- intercept form

with slope m = - 3

given a line with slope m then the slope of a line perpendicular to it is

`m_(perpendicular)` = -
`(1)/(m)` = -
`(1)/(-3)` =
`(1)/(3)` , then

y =
`(1)/(3)` x + c ← is the partial equation

to find c substitute P (6, 10 ) into the partial equation

10 =
`(1)/(3)` (6) + c = 2 + c ( subtract 2 from both sides )

8 = c

y =
`(1)/(3)` x + 8 ← equation of perpendicular line