# Solving equations in SageMath

Hey! I'm new to sage and I'm trying to see how complex are the calculations that I can express in SageMath. So I considered the following example, the comparison between the likelihood of two normals with different means but same variance:

$$\dfrac{1}{\sigma\sqrt{2\pi} } \exp{\left(-\dfrac{(x-\mu_1)^2}{2\sigma^2}\right)} = \dfrac{1}{\sigma\sqrt{2\pi} } \exp{\left(-\dfrac{(x-\mu_2)^2}{2\sigma^2}\right)}$$

I know that the answer must be $\mu_1 = \mu_2$. Can I solve it with SageMath? I tried:

x, mu1, mu2, sigma = var('x, mu1, mu2, sigma')
exp((0.5/sigma)*(x-mu1)^2)==exp((0.5/sigma)*(x-mu2)^2)


and

solve(exp((0.5/sigma)*(x-mu1)^2)==exp((0.5/sigma)*(x-mu2)^2))


but that doesn't seem to work. Is it possible to do so in sagemath? Thanks!!

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You can solve for mu1:

var('x, mu1, mu2, sigma')
result=solve(exp((0.5/sigma)*(x-mu1)^2)==exp((0.5/sigma)*(x-mu2)^2),mu1)
show(result)

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By the way, what if I wanted to solve for two variables, meaning I wanted to check if for different $\sigma_1$ and $\sigma_2$, I tried the following:

var('x, mu1, mu2, sigma1, sigma2')
result = solve((1/sigma1^2)*(x-mu1)^2 + log(sigma1)==(1/sigma2^2)*(x-mu2)^2+log(sigma2), mu1, sigma1)


But it didnt't work out. How can I define for several variables the solution?

1

Usually, you need a second equation to solve for two variables.