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?

1 Answer
Nov 6, 2015

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.