How do you solve the system of equations by graphing and then classify the system as consistent or inconsistent #12x + 4y = -4# and #2x -y =6#?

To graph each line we need to find two points of solution for the equation, map the points then draw a line through the two points:

Equation 1

For #x = -2#

#(12 xx -2) + 4y = -4#

#-24 + 4y = -4#

#color(red)(24) - 24 + 4y = color(red)(24) - 4#

#0 + 4y = 20#

#4y = 20#

#(4y)/color(red)(4) = 20/color(red)(4)#

#y = 5# or #(-2, 5)#

For #x = 0#

#(12 xx 0) + 4y = -4#

#0 + 4y = -4#

#4y = -4#

#(4y)/color(red)(4) = -4/color(red)(4)#

#y = -1# or #(0, -1)#

graph{((x+2)^2+(y-5)^2-0.05)(x^2+(y+1)^2-0.05)(12x+4y+4)=0 [-15, 15, -7.5, 7.5]}

Equation 2

For #x = 0#

#(2 xx 0) - y = 6#

#0 - y = 6#

#-y = 6#

#color(red)(-1) xx -y = color(red)(-1) xx 6#

#y = -6# or #(0, -6)#

For #y = 0#

#2x - 0 = 6#

#2x = 6#

#(2x)/color(red)(2) = 6/color(red)(2)#

#x = 3# or #(3, 0)#

graph{(2x-y-6)((x-3)^2+y^2-0.05)(x^2+(y+6)^2-0.05)(12x+4y+4)=0 [-15, 15, -7.5, 7.5]}

We can see the lines intersect at #(1, -4)# for the solution to the problem.

graph{(2x-y-6)((x-1)^2+(y+4)^2-0.0125)(12x+4y+4)=0 [-8, 8, -7, 1]}

Because there is at least one solution the system of equations is consistent.