Linear algebraConcept reference

Unique solutions

A real linear system has a unique solution exactly when it is consistent and has no free variables.

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01A unique solution is the only real assignment satisfying every equation#

A unique solution is the only real assignment satisfying every equation. Here $x=2$ and $y=-1$ are both fixed.

Now compare $x+y=2$ . Choosing $y=0$ gives $x=2$ ; choosing $y=1$ gives $x=1$ . Both pairs work.

The whole family is

(x,y)=(2-t,t)

$t$ can be any real number. Consistency permits free choices.

For a real linear system, first check consistency. Then check for free variables. A consistent system is unique exactly when no variable is free.

In reduced form, every variable column must have a pivot. A row such as $0=1$ rules out any solution, even if the variable columns have pivots.

Check your reasoning

A reduced system is $x-2y=3$ , $0=0$ . How many real solutions?

Show answer and explanation

Infinitely many.

Choose any $y$ ; then $x=3+2y$ .

The system $x=3$ , $y=-2$ , $0=0$ is consistent and has no free variable. The zero row adds no restriction; the solution is still unique.

Repeated equations impose the same restriction. They leave the same variables free.

Check your reasoning

Real $a,b$ satisfy $a+b=7$ , $b=2$ . Unique solution?

Show answer and explanation

Yes: $(5,2)$ .

$b=2$ forces $a=5$ .

Check your reasoning

Lee calls $(1,3)$ the unique solution of $x+y=4$ . What refutes this?

Show answer and explanation

$(2,2)$ also solves it.

Both pairs satisfy $x+y=4$ .

Key takeaway