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]}