How do you solve #2 x + 3 y = -7# and #2 x - 7 y = 7# using substitution?

Arrange the first equation:

#2x = -3y - 7#

Now put this into the second equation:

# -3y - 7 - 7y = 7#

# -10y - 7 = 7#

# -10 y = 14#

#y = -14/10# or

#y = -7/5#

Put this into the first original equation

# 2x + 3times(-7/5) = -7#

#2x -21/5 = -7#

#2x = 21/5 - 7#

#2x = (21-35)/5#

#2x = -14/5#

#x = -14/10#

#x = -7/5#

Your solution is #x=-7/5# and #y=-7/5#

Step 1) We can solve both equations for #2x# which is a common term:

Equation 1)

#2x + 3y = -7#

#2x + 3y - color(red)(3y) = -7 - color(red)(3y)#

#2x + 0 = -7 - 3y#

#2x = -7 - 3y#

Equation 2)

#2x - 7y = 7#

#2x - 7y + color(red)(7y) = 7 + color(red)(7y)#

#2x - 0 = 7 + 7y#

#2x = 7 + 7y#

Step 2) Equate the right sides of each equation and solve for #y#:

#-7 - 3y = 7 + 7y#

#-color(blue)(7) - 7 - 3y + color(red)(3y) = -color(blue)(7) + 7 + 7y + color(red)(3y)#

#-14 - 0 = 0 + (7 + color(red)(3))y#

#-14 = 10y#

#-14/color(red)(10) = (10y)/color(red)(10)#

#-7/5 = (color(red)(cancel(color(black)(10)))y)/cancel(color(red)(10))#

#-7/5 = y#

#y = -7/5#

Step 3) Substitute #-7/5# for #y# in the solution to either equation in Step 1 and calculate #x#. I will choose the second equation:

#2x = 7 + 7y# becomes:

#2x = 7 + (7 * -7/5)#

#2x = 7 + (-49/5)#

#2x = (7 xx 5/5) - 49/5#

#2x = 35/5 - 49/5#

#2x = -14/5#

#color(red)(1/2) xx 2x = color(red)(1/2) xx -14/5#

#1x = -14/10#

#x = -7/5#

The solution is: #x = -7/5# and #y = -7/5# or #(-7/5, -7/5)#