To find the y-intercept of the line, we can first find the equation of the line using the given slope. Then, we can find the x-coordinate of the point where the line intersects one of the sides of the unit square. Finally, we can substitute this x-coordinate into the equation of the line to find the y-intercept.
Let's start by finding the equation of the line with a slope of 6. The slope-intercept form of the equation of a line is given by y = mx + b, where m is the slope and b is the y-intercept.
Since the slope is given as 6, we have y = 6x + b.
Now, let's find the x-coordinate of the point where the line intersects one of the sides of the unit square. Since the line bisects the area of the square, its midpoint must lie on the line. The coordinates of the midpoint can be found by taking the average of the coordinates of the opposite corners of the square.
The opposite corners of the square are (1,0) and (0,1). Their average gives us the coordinates of the midpoint:
Midpoint x-coordinate = (1 + 0) / 2 = 0.5
Next, substitute this x-coordinate (0.5) into the equation of the line to solve for the y-coordinate of the midpoint:
y = 6(0.5) + b
y = 3 + b
Since the midpoint lies on the line, we can substitute the coordinates of the midpoint (0.5, y) into the equation of the line:
y = 6(0.5) + b
y = 3 + b
Now, we have two equations with two variables (y and b):
(1) y = 3 + b
(2) y = 6(0.5) + b
Solving these equations simultaneously will give us the values of y and b. Subtracting equation (1) from equation (2), we get:
0 = 3 - 6(0.5)
0 = 3 - 3
0 = 0
The equation is satisfied, which means the system of equations is dependent, and there are infinitely many solutions. This implies that the y-intercept (b) can be any value.
Therefore, the y-intercept of the line is not uniquely determined and can take on any real number value.