Question

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?

Answer 1

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]}