Consistency in Linear Equations: The Relationship Between Coefficients c and d

cmajor47
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Homework Statement


Suppose the system below is consistent for all possible values of f and g. What can you say about the coefficients c and d?


Homework Equations


x1 + 3x2 = f
cx1 + dx2 = g

The Attempt at a Solution


I really just have no idea what this says about c and d.
 
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Try solving the equations. Multiply the first one by c and subtract them and solve for x2. What could go wrong with this procedure?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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