proof and rules of inferences

**braddy** · Sep 3rd 2006, 02:55 PM

Hi,
Please can someone help me with this problem.

show that a,b,c are real numbers and a#0, then there is a unique solution of the equation ax+b=c.

the uniqueness of the solution is my problem.
Thank you
B

**ThePerfectHacker** · Sep 3rd 2006, 03:28 PM

Originally Posted by braddy

Hi,
Please can someone help me with this problem.

show that a,b,c are real numbers and a#0, then there is a unique solution of the equation ax+b=c.

the uniqueness of the solution is my problem.
Thank you
B

Okay.
If,
$ax+b=c$
There exists preciesly one such number $-b$ such as $b+(-b)=0$ because the real numbers form a group under addition.
Thus,
$(ax+b)+(-b)=c+(-b)$
Since addition is associative we have,
$ax+(b-b)=c-b$
Thus,
$ax+0=c-b$
Since,
$0$ is an identity element we have,
$ax=b$
Since, $a\not = 0$ it is an element of the real numbers under multiplication and form a group. Thus, there exist a unique, $1/a$ such as, $a(1/a)=1$ .
Thus,
$(1/a)(ax)=(1/a)(c-b)$
Since, multiplication is associative we have,
$((1/a)a)x=1/a(c-b)$
Thus,
$1x=1/a(c-b)$
Since 1 is the multiplicative identity element we have,
$x=1/a(c-b)$

**Plato** · Sep 3rd 2006, 04:38 PM

Here is the way that I would expect this to be done.
Suppose that each of p and q is a solution to the equation $ax + b = c,\quad a \not= 0$ .
Then
$\begin{array}{rcl} ap + b & = & c \\ aq + b & = & c \\ ap + b & = & aq + b \\ ap & = & aq \\ p & = & q \\ \end{array}$

**ThePerfectHacker** · Sep 3rd 2006, 04:45 PM

Originally Posted by Plato

Here is the way that I would expect this to be done.
Suppose that each of p and q is a solution to the equation $ax + b = c,\quad a \not= 0$ .
Then
$\begin{array}{rcl} ap + b & = & c \\ aq + b & = & c \\ ap + b & = & aq + b \\ ap & = & aq \\ p & = & q \\ \end{array}$

Yes but you do not show that a solution exists

**Plato** · Sep 3rd 2006, 05:00 PM

No, you are correct!
But you showed a solution existed.
I simplified its uniqueness.

**braddy** · Sep 4th 2006, 10:24 PM

Originally Posted by Plato

No, you are correct!
But you showed a solution existed.
I simplified its uniqueness.

Thank you forr your Help!

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