# A line segment is bisected by a line with the equation # - y + 3 x = 1 #. If one end of the line segment is at #(6 ,3 )#, where is the other end?

Using the point slope method, find the equation for this line:

The midpoint's coordinates can be found by using:

Then:

The opposite endpoint's coordinates are:

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To find the other end of the line segment bisected by the line -y + 3x = 1, given that one end is at (6, 3), we first need to find the equation of the line segment represented by the bisecting line. Then, we can find the intersection point of this line with the given line segment to locate the other end.

**Find the equation of the bisecting line:**

Given the equation of the bisecting line is -y + 3x = 1, we rearrange it into slope-intercept form (y = mx + b):

-y + 3x = 1 => y = 3x - 1

So, the slope of the bisecting line is 3.

**Find the midpoint of the line segment:**

Given one end of the line segment is at (6, 3), we'll use the midpoint formula to find the midpoint. Let's denote the other end as (a, b), then the midpoint (M) is:

M = ((6 + a)/2, (3 + b)/2)

**Use the midpoint and slope to find the other end:**

Since the bisecting line goes through the midpoint (M) and has a slope of 3, we can use the point-slope form to find the equation of this line:

y - y₁ = m(x - x₁)

Substituting the midpoint coordinates (6 + a)/2 and (3 + b)/2 for (x₁, y₁), we have:

y - (3 + b)/2 = 3(x - (6 + a)/2)

Now, we substitute the point (6, 3) into this equation to solve for the other end (a, b).

3 = 3(6 - (6 + a)/2) 3 = 18 - 3(a + 6)/2 3 = 18 - (3/2)(a + 6) 3 = 18 - (3/2)a - 9 9 = (3/2)a a = 6

Substitute a = 6 into the equation to find b:

b - 3/2 = 3(6 - (6 + 6)/2) b - 3/2 = 3(6 - 6) b - 3/2 = 0 b = 3/2

Therefore, the other end of the line segment is at (6, 3/2).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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