How do solve the following linear system?: # x+2y=1 , -3x -8y = -9 #?
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#x+2y=1#
#x=1-2y# -----(1)
#-3x-8y =-9#
#3x + 8y = 9# -----(2)
Substituting (1) into (2):
#3(1-2y) + 8y = 9#
# 2y +3 = 9#
# y = 3#
Substituting y=3 into (1):
#x= 1-2(3)= -5#
See the entire solution process below:
Step 1) Solve the first equation for #x#:
#x + 2y = 1#
#x + 2y - color(red)(2y) = 1 - color(red)(2y)#
#x + 0 = 1 - 2y#
#x = 1 - 2y#
Step 2) Subsitute #1 - 2y# for #x# in the second equation and solve for #y#:
#-3x - 8y = -9# becomes:
#-3(1 - 2y) - 8y = -9#
#(-3 * 1) - (-3 * 2y) - 8y = -9#
#-3 - (-6y) - 8y = -9#
#-3 + 6y - 8y = -9#
#-3 + (6 - 8)y = -9#
#-3 - 2y = -9#
#color(red)(3) - 3 - 2y = color(red)(3) - 9#
#0 - 2y = -6#
#-2y = -6#
#(-2y)/color(red)(-2) = -6/color(red)(-2)#
#(color(red)(cancel(color(black)(-2)))y)/cancel(color(red)(-2)) = 3#
#y = 3#
Step 3) Substitute #3# for #y# in the solution to the first equation at the end of Step 1 and calculate #x#:
#x = 1 - 2y# becomes:
#x = 1 - (2 * 3)#
#x = 1 - 6#
#x = -5#
The solution is: #x = -5# and #y = 3# or #(-5, 3)#
