How do I solve systems of equations with multiple variables?

NINHARDCOREFAN
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How do I solve this:
.5x+.5y=.5
.5x^2+.5y^2=1/3
 
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working with multiple equations to find an equal multiple of variables requires solving and subsitution.

First, pick x or y, and one equation, and solve the chosen equation for the chosen variable.

The term that x is equal to can be substituted in for x in the leftover equation to find a specific y, which can then be substituted into either equation to find x.
 

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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