Determining if a linear system has no, unique or infinitely many solns

I'm having trouble with these questions:

If given a linear system:

$\displaystyle (m-2)x+y=2$

and

$\displaystyle mx+2y=k$

For which values of m and k (unique values or certain sets of values) are the following true:

1: The system has unique solutions

2: The system has no unique solutions

3: There are infinitely many solutions

Or, consider the system of three equations:

$\displaystyle 2x+ay-z=0$

$\displaystyle 3x+4y-(a+1)z=13$

$\displaystyle 10x+8y+(a-4)z=26$

For what values of a are the same three conditions above satisfied?

Also, what techniques are used to do these questions in general?

Thanks for all the help in advance

Re: Determining if a linear system has no, unique or infinitely many solns

The number of solutions is intimately related to the determinant of the corresponding matrix.

The determinant of your first system is:

$\displaystyle \begin{vmatrix}(m -2)&1\\m&2\end{vmatrix}$

And this equals $\displaystyle m - 4$

When the determinant is NOT equal to zero, you'll have a unique solution. So $\displaystyle m \neq 4$ gives a unique solution.

When m equals 4, then we must consider values of k. m = 4 makes the system:

$\displaystyle 2x + y = 2$

$\displaystyle 2x + y = k$, which might be easier to analyse without matrices. If k is not 2, then there is no solution. If k = 2, there are infinite solutions.

Re: Determining if a linear system has no, unique or infinitely many solns

Quote:

Originally Posted by

**Istafa** I'm having trouble with these questions:

If given a linear system:

$\displaystyle (m-2)x+y=2$

and

$\displaystyle mx+2y=k$

For which values of m and k (unique values or certain sets of values) are the following true:

1: The system has unique solutions

2: The system has no unique solutions

3: There are infinitely many solutions

...

I am not sure if you are allowed to use determinants. If so:

1. Calculate the main-determinant:

$\displaystyle D=\left|\begin{array}{cc}m-2&1 \\ m&2\end{array}\right|=2m-4-m=m-4$

That means you'll get a unique solution if $\displaystyle m\ne4$

2. The solutions are calculated by the quotients of the determinants:

$\displaystyle x = \frac{D_x}D$ and $\displaystyle y = \frac{D_y}D$

with

$\displaystyle D_x= \left|\begin{array}{cc}2&1 \\ k&2\end{array}\right|= 4-k$

$\displaystyle D_y= \left|\begin{array}{cc}m-2&2 \\ m&k\end{array}\right|= m(k-2)-2k$

3. If m = 4 **and **k = 4 you'll get infinitely many solutions.

If m = 4 **and **$\displaystyle k \ne 4$ then the system of equations doesn't have a solution.

Re: Determining if a linear system has no, unique or infinitely many solns

Another approach is to think graphically. You have two straight lines represented by your two linear equations. No solutions means the lines are parallel but not the same line. One solution means the lines are not parallel. Infinitely many solutions means the lines are the same lines.