Question

How do you determine whether a linear system has one solution, many solutions, or no solution when given 2x+3y= -12 and 2x-y = -4?

Answer 1

If the gradients are different in a 2 equation linear system then 1 solution

If they are parallel then no solution. They will not cross.

Explanation:

In your case it is given that:
`2x+3y=-12`..................................(1)
`2x-y=-4`.......................................(2)

Write as:

`y=-2/3 x -4`...........................(`1_a`)
`y= 2x+4`....................................(`2_a`)

The gradient of (`1_a`) is negative so its plot has a downward slope when progressing from left to right.

The gradient of (`2_a`) is positive so its plot has an upward slope as you progress from left to right.

As there is no restraint on the magnitude of `x` we have `oo<=x<=oo` so at some point they will cross and then carry on. This will produce just 1 solution.