Question

How do you determine whether a linear system has one solution, many solutions, or no solution when given 7x+5y= -12 and 3x-4y=1?

Answer 1

In this instance, one solution.

Explanation:

`7x+5y=-12 -> " " y=-7/5x - 12/5` ...... ( 1 )

`3x-4y=1 -> " " y=3/4x-1/4`..................( 2 )

These two straight plots have different gradients (coefficient of `x`). That means that eventually they will cross at some as yet not calculated point. A bit like the junction of two infinitely long roads in space, which for simplicity of explanation, one of which is the equivalent of North/South and the other the equivalent of East/West. They will only cross at one point. For them to cross at more than one point curves would be involved. They would both have to be spatially placed appropriately. For some curve types they could miss each other.

If they had the same gradient they would be parallel to each other. In this context the only way that they would "have contact" with each other is if one was superimposed with the other. In which case you would be repeating the equation.