Further Investigations
There is another way of telling whether a system of two linear equations in two unknowns is consistent or inconsistent, or whether the equations are dependent, without taking the time to graph each equation. It can be shown that any system of the form
Has one and only one solution if
That it has no solution if
And that it has infinitely many solutions if
For each of the following systems, determine whether the system is consistent, the system is inconsistent, or the equations are dependent.
(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(a)
Whether the system is consistent, the system is inconsistent, or the equations are dependent.
Approach:
The system of the form,
i) Has one and only one solution if,
ii) That it has no solution if,
iii) And that it has infinitely many solutions if,
(b)
Whether the system is consistent, the system is inconsistent, or the equations are dependent.
(c)
Whether the system is consistent, the system is inconsistent, or the equations are dependent.
(d)
Whether the system is consistent, the system is inconsistent, or the equations are dependent.
(e)
Whether the system is consistent, the system is inconsistent, or the equations are dependent.
(f)
Whether the system is consistent, the system is inconsistent, or the equations are dependent.
(g)
Whether the system is consistent, the system is inconsistent, or the equations are dependent.
(h)
Whether the system is consistent, the system is inconsistent, or the equations are dependent.
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