A line 𝑙l is such that its segment between the lines 𝑥−𝑦+2=0x−y+2=0 and 𝑥+𝑦−1=0x+y−1=0 is internally bisected at the point (1,1.5)(1,1.5). What is the equation of the line 𝑙l?

The given lines are x - y + 2 = 0 and x + y - 1 = 0. Let's first find their intersection point.

To find the intersection point, we can solve the two equations simultaneously.

Adding the two equations, we get: x - y + 2 + x + y - 1 = 0 => 2x + 1 = 0 => x = -1/2

Substituting x = -1/2 in the first equation, we get: -1/2 - y + 2 = 0 => y = 2 - 1/2 = 3/2

So, the intersection point of the two lines is (-1/2, 3/2).

Now, the line segment between the two lines is bisected at the point (1, 1.5). This means that the point (1, 1.5) is the midpoint of the line segment.

The midpoint of two points (x1, y1) and (x2, y2) is given by ((x1 + x2)/2, (y1 + y2)/2).

So, if (-1/2, 3/2) and (x, y) are the endpoints of the line segment, then its midpoint is ((-1/2 + x)/2, (3/2 + y)/2) = (1, 1.5).

Solving the equations (-1/2 + x)/2 = 1 and (3/2 + y)/2 = 1.5, we get x = 5/2 and y = 3/2.

So, the other endpoint of the line segment is (5/2, 3/2).

The equation of a line passing through two points (x1, y1) and (x2, y2) is given by (y - y1) = ((y2 - y1)/(x2 - x1))(x - x1).

Substituting the values of the points (-1/2, 3/2) and (5/2, 3/2) in this equation, we get the equation of the line as y - 3/2 = 0*(x + 1/2), which simplifies to y = 3/2.

So, the equation of the line l is y = 3/2.