A line segment is bisected by a line with the equation # 4 y - 3 x = 2 #. If one end of the line segment is at #( 2 , 1 )#, where is the other end?

Question

A line segment is bisected by a line with the equation # 4  y  - 3  x   =   2  #.  If one end of the line segment is at #(    2 ,  1  )#, where is the other end?

Sophia Alfred · Accepted Answer

#(1 1/25,2 7/25)# #4y-3x=2#, #4y=3x+2# #y=3/4x+1/2# #->a# gradient of the other line, #m = -1/(3/4)=-4/3# The equation of the line thru #(2,1)#, #y-y_1=m(x-x_1)# #y-1=-4/3(x-2)# #y=-4/3x+8/3+1# #y=-4/3x+11/3# replacing y from #a# #3/4x+1/2=-4/3x+11/3# #3/4x+4/3x=11/3-1/2# #9/12x+16/12x=22/6-3/6# #25/12x=19/6# #x=19/6*12/25=38/25# #y=3/4(38/25)+1/2# #y=57/50+25/50=82/50=41/25# #(38/25,41/25)# is a midpoint of the line. The other end of the line, #(x,y)#. #(x+2)/2=38/25# #x=38/25*2-2# #x=76/25-50/25=26/25=1 1/25# #(y+1)/2=41/25# #y=41/25*2-1# #y=82/25-25/25=57/25=2 7/25# The other end #(1 1/25,2 7/25)#

Sadie Adams · Answer

undefined To find the other end of the line segment bisected by the line with the equation $4y - 3x = 2$ and one end at (2, 1), we can first find the midpoint of the line segment using the given point and then use the midpoint formula to find the other end.

The midpoint formula is $\left(\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}\right)$. Given the point (2, 1) and assuming the other end is (x, y), we can set up the midpoint formula as:

$\left(\frac{{2 + x}}{2}, \frac{{1 + y}}{2}\right)$

Now, since the line bisects the line segment, it passes through the midpoint. We can substitute the midpoint coordinates into the equation of the line $4y - 3x = 2$ and solve for x and y.

Substitute the midpoint coordinates into the equation:

$4\left(\frac{{1 + y}}{2}\right) - 3\left(\frac{{2 + x}}{2}\right) = 2$

Simplify and solve for x and y:

$2(1 + y) - 3(2 + x) = 2$

$2 + 2y - 6 - 3x = 2$

$-4 - 3x + 2y = 0$

$3x - 2y = -4$

Now, plug in the given point (2, 1) to find x and y:

$3(2) - 2(1) = -4$

$6 - 2 = -4$

$4 = -4$ (This is not possible, indicating that there is no solution with the given information.)

Therefore, based on the given information, it's not possible to determine the exact coordinates of the other end of the line segment.