How do you determine whether a linear system has one solution, many solutions, or no solution when given 6x +y= -6 and 4x+3y= 17?

1 Answer
Oct 13, 2015

In this case we can reformulate the equations as two slope intercept equations describing lines of different slope and therefore one solution.

Explanation:

Slope intercept form of the equation of a line is:

#y = mx + c#

where #m# is the slope and #c# the intercept.

Starting with #6x+y = -6#, subtract #6x# from both sides to get:

#y = -6x-6#

This is a line with slope #-6# and intercept #-6#.

Starting with #4x+3y=17#, first subtract #4x# from both sides to get:

#3y = -4x+17#

Then divide both sides by #3# to get:

#y = -4/3x + 17/3#

This is a line with slope #-4/3# and intercept #17/3#

Since the slopes of the two lines are different, the lines intersect at exactly one point.

graph{(6x+y+6)(4x+3y-17) = 0 [-20.78, 19.22, -4.16, 15.84]}