Question

How do you determine the solution in terms of a System of Linear Equations for -2x + 3y =5 and ax - y = 1?

Answer 1

This system has:

• No solution if `a=2/3`
• One solution : `{(x=(-8)/(2-3a)), (y=(-2-5a)/(2-3a)):}` if `a!=2/3`

Explanation:

To find the connection between value of a parameter `a` and the number of solutions of a linear system you can use the Cramer's Rule.

It can be written as follows:

Let there be a system of 2 linear equations:

`{(a_1x+b_1y=c_1),(a_2x+b_2y=c_2):}`

Let

`W=a_1*b_2-b_1*a_2`

`W_x=c_1b_2-b_1c_2`

`W_y=a_1c_2-c_1a_2`

Then the system has:

• One solution `{(x=W_x/W),(y=W_y/W):} iff W!=0`
• No solutions `iff W=0` and `W_x!=0` or `W_y!=0`
• Infinitely many solutions `iff W=0` and `W_x=0` and `W_y=0`

This rule can be expanded for any system of `n` equations with `n` variables, then you have `W` and for every variable `x_i` you have a corresponding determinant `W_(x_i)`, and the rule says that:

1. If `W!=0` system has exactly one solution: `x_i=(W_{x_i})/W` for `1<=i<=n`

2. If `W=0` and any of `W_{x_i}` is not zero, then system has no solutions

3. If `W=0` and all `W_{x_i}` are zeros, then the system has infinitely many solutions.