3 unknowns and 1 equation
Find the values of a and b if,
$$\frac{a}{x+3}-\frac{b}{x+2}=\frac{3x-2}{(x+3)(x+2)}$$
The answer shows to simultaneously solve these two equations:
$a-b=3$
$2a-3b=-2$
But I have no idea how in the world to get these two equations. Any help would be appreciated.
$\endgroup$ 12 Answers
$\begingroup$The standard method for this is called partial fraction decomposition.
Basically you’re finding rational functions with denominators of lower degree by using the method of equating coefficients which is itself a useful and powerful technique.
Ets1331 showed how to do it in this case.
$\endgroup$ $\begingroup$We know that this equation is true for any value of $x$. This is how you get more than one equation: you plug in more than one value of $x$.
We have that $\frac{a}{x+3} - \frac{b}{x+2} = \frac{a(x+2) - b(x+3)}{(x+3)(x+2)}$, so we know $a(x+2)-b(x+3) = 3x-2 \implies (a-b)x + (2a - 3b) = 3x-2$. Because this should work for any value of $x$, we have that $a - b = 3$ and $2a - 3b = -2$.
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