# Solving this equation with 3 variables.

I came across this equation.

$m = (m + 1/2)x + (b-1)$

The goal is to solve for m and b.

I want to make sure I understand the very last algebraic step required. Since the two sides are equal, you force the first term to equal 0, thereby having m equal the 2nd term. ie: You set m = -1/2 and then m must equal (b-1) You subsequently sub in that m=-1/2 to solve for b.

It makes sense once I see it, but am not sure I'd think of that. Is there a name for this technique? Is this the only possible solution? What is the general strategy?

You intentionally pick m=-1/2 to conveniently get rid of that m ?
To solve for 2 variables, you MUST get rid of the 3rd (x)?

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You can't "solve" a single equation in 3 unknowns. But you may be able to express each variable in terms of the other two (in two different equations). That is, you could write $m = \text{(expression involving only }x\text{ and }b\text{)}$, $x = \text{(expression involving only }m\text{ and }b\text{)}$, and $b = \text{(expression involving only }m\text{ and }x\text{)}$. Is that what you mean?

If so, the easiest way is to clear parentheses by distributing the multiplication, then use the result three different ways to collect all terms involving the desired variable on one side, factor it out of that side, then divide both sides by the other factor:

$m = mx + \frac{1}{2}x + b - 1$

For $m$:

$m - mx = \frac{1}{2}x + b - 1 \Rightarrow m(1 - x) = \frac{1}{2}x + b - 1 \Rightarrow \boxed{m = (\frac{1}{2}x + b - 1)/(1 - x)}$

For $x$:

$mx + \frac{1}{2}x = m - b + 1 \Rightarrow x(m + \frac{1}{2}) = m - b + 1 \Rightarrow \boxed{x = (m - b + 1) / (m + \frac{1}{2})}$

For $b$:

$\boxed{b = m - mx - \frac{1}{2}x + 1}$

You can factor various parts of the final expressions, but that's a matter of your own taste.

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$m = (m + 1/2)x + (b-1)$ So, by "equating the coefficients" you think of the left side as $0x + m$. this gives you $0x + m = (m + 1/2)x + (b-1)$ That means $m + 1/2 = 0$ --> m=-1/2. m=b-1 --> But, m=-1/2 --> -1/2=b-1 --> b=1/2 ... Solution: m=-1/2 & b=1/2. –  JackOfAll Jan 24 '14 at 2:05

Here is what I was told:

$m = (m + 1/2)x + (b-1)$

So, by "equating the coefficients" you think of the left side as $0x + m$.

This gives you $$0x + m = (m + 1/2)x + (b-1)$$

Matching up the terms, that means $0 = m + 1/2$ and $m=b-1$

So m=-1/2

Now, sub m into $m=b-1$

-1/2=b-1

b=1/2

... Solution: m=-1/2 & b=1/2.

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