The line L with equation y=mx+n passes through the point (1,1) and is perpendicular to x+3y=1. Find the value of n.

the slope of L is 3, so use the point-slope form of the line.

L: y-1 = 3(x-1)
Now use that to find n.

To find the value of n, we first need to determine the slope (m) of the line L. Since line L is perpendicular to the line x + 3y = 1, the slopes of these two lines are negative reciprocals of each other.

The equation x + 3y = 1 can be rewritten in slope-intercept form (y = mx + b) by solving for y:
3y = -x + 1
y = (-1/3)x + 1/3

Comparing this equation to y = mx + n, we can see that the slope (m) of the line x + 3y = 1 is -1/3.

Since line L is perpendicular to x + 3y = 1, its slope will be the negative reciprocal of -1/3. We can find this slope by taking the negative reciprocal of -1/3:

m = -1/(-1/3) = 3

Now, we have the slope (m = 3) of line L and a point it passes through (1, 1). We can use the point-slope form of a linear equation to find the equation of line L:

y - y1 = m(x - x1)

Plugging in the coordinates of the point (1, 1):

y - 1 = 3(x - 1)

Expanding the equation:

y - 1 = 3x - 3

Now, we can rewrite this equation in the slope-intercept form (y = mx + n) by solving for y:

y = 3x - 2

By comparing this equation to y = mx + n, we can see that the value of n is -2.

Therefore, the value of n is -2.