#color(red)("Using shortcut method")#

Move #5 2/7# to the other side

#k=color(brown)(2 27/70)color(green)(-5 2/7)#

By reversing the numbers I can subtract the lesser value from the greater. However, I have to multiply the whole by (-1) to maintain the original values.

Write as #k= -1xx(color(green)(4 9/7)color(brown)(-2 (2.7)/7)) = -2 (6.3)/7=-2 63/70 = -2 9/10#

#color(blue)(" "k= -2 9/10)#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(red)("Using first principles method")#

I have elected to show this process in a lot of detail

Given:#" "5 2/7+k=2 27/70#

Subtract #5 2/7# from both sides

#k=2 27/70-5 2/7#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(blue)("Consider "2 27/70)#

Write as #2+27/70#

Multiply the 2 by 1 but in the form #1=70/70#

#(2/1xx70/70)+27/70" "->" "140/70+27/70 color(blue)(= 167/70)#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(blue)("Consider "5 2/7)#

Write as #5+2/7#

Multiply 5 by 1 but in the form #1=7/7#

#(5/1xx7/7)+2/7" "->" "35/7+2/7 = 37/7#

Multiply by 1 but in the form #10/10#

#37/7xx10/10color(blue)( = 370/70)#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(blue)("Putting it all together")#

#color(brown)(" "k=2 27/70-5 2/7" "->" "color(green)(k=167/70-370/70)#

#color(blue)(" "k=-203/70 =-2 9/10)#