How do you solve the system of equations by graphing and then classify the system as consistent or inconsistent #6x - 7y = 49# and #7y - 6x = -49#?

First, we need to graph each line by finding two solutions for each equation, plotting the solution points and then drawing a line through the points.

Equation 1:

• First Point: #x = 0#

#(6 xx 0) - 7y = 49#

#0 - 7y = 49#

#(-7y)/color(red)(-7) = 49/color(red)(-7)#

#y = -7# or #(0, -7)#

• Second Point: #x = 7#

#(6 xx 7) - 7y = 49#

#42 - 7y = 49#

#42 - color(red)(42) - 7y = 49 - color(red)(42)#

#0 - 7y = 7#

#(-7y)/color(red)(-7) = 7/color(red)(-7)#

#y = -1# or #(7, -1)#

-First Graph:

graph{(x^2 + (y+7)^2 - 0.075)((x - 7)^2 + (y+1)^2 - 0.075)(6x - 7y - 49) = 0 [-20, 20, -10, 10]}

Equation 2:

• First Point: #x = 0#

#7y - (6 xx 0) = -49#

#7y - 0 = -49#

#(7y)/color(red)(7) = -49/color(red)(7)#

#y = -7# or #(0, -7)#

• Second Point: #x = 7#

#7y - (6 xx 7) = -49#

#7y - 42 = -49#

#7y - 42 + color(red)(42) = -49 + color(red)(42)#

#7y - 0 = -7#

#(7y)/color(red)(7) = -7/color(red)(7)#

#y = -1# or #(7, -1)#

-First and Second Graph:

graph{(x^2 + (y+7)^2 - 0.075)((x - 7)^2 + (y+1)^2 - 0.075)(6x - 7y - 49) = 0 [-20, 20, -10, 10]}

Solutoin:

As we can see from the graph, both equations represent the same line. Therefore, there are an infinite number of solutions.

The lines are Consistent because there is at least one solution.