Questions tagged [systems-of-equations]

This tag indicates that several equations (of some type) must all hold. Do not use alone! Use in conjunction with (linear-algebra), (polynomials), (pde), (differential-equations), (inequalities) or another tag that describes the nature of the equations being considered.

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Is it possible to have three real numbers that have both their sum and product equal to $1$?

I have to solve $x+y+z=1$ and $xyz=1$ for a set of $(x, y, z)$. Are there any such real numbers? Edit : What if $x+y+z=xyz=r$, $r$ being an arbitrary real number. Will it still be possible to find ...
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Solution to the equation of a polynomial raised to the power of a polynomial.

The problem at hand is, find the solutions of $x$ in the following equation: $$(x^2−7x+11)^{x^2−7x+6}=1$$ My friend who gave me this questions, told me that you can find $6$ solutions without ...
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How does Cramer's rule work?

I know Cramer's rule works for 3 linear equations. I know all steps to get solutions. But I don't know why (how) Cramer's rule gives us solutions? Why do we get $x=\frac{\Delta_1}\Delta$ and $y$ and ...
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Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE's.

The original ODE I had was $$\frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$$ with $y(0)=3$ and $y'(0)=1$. Now I can solve this by hand and obtain that $y(1) = 14.82789927$. However I wish to use the 4th order ...
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Difference between least squares and minimum norm solution

Consider a linear system of equations $Ax = b$. If the system is overdetermined, the least squares (approximate) solution minimizes $||b - Ax||^2$. Some source sources also mention $||b - Ax||$. If ...
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If $2^x=3^y=6^{-z}$ then prove that:$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0$

If $$2^x=3^y=6^{-z}$$ and $x,y,z \neq 0$ then prove that:$$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0$$ I have tried starting with taking logartithms, but that gives just some more equations. Any ...
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Solving $x_1+x_2=x_3^2, x_2+x_3=x_4^2, x_3+x_4=x_5^2,x_4+x_5=x_1^2, x_5+x_1=x_2^2$ in reals

find answers of this system of equations in real numbers$$\left\{ \begin{array}{c} x_1+x_2=x_3^2 \\ x_2+x_3=x_4^2 \\ x_3+x_4=x_5^2 \\ x_4+x_5=x_1^2 \\ x_5+x_1=x_2^2 \end{array} \right.$$ ...
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Pell number factorization and divisibility question

In a problem I’m working on, I have positive integers $a,b,c,d$ satisfying $$(ab)^2-2(cd)^2=1. \tag{1}$$ (So evidently $cd$ is a Pell number, and $ab$ is its companion.) Furthermore, say the ...
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Given a set of Hermitian matrices $\{A_i\}$, is there a simple way to check if there exists a vector $c$ such that for all $i$: $$c^* A_i c = 0?$$ Namely, when can the quadratic forms defined by the ...
$$\begin{cases} 2x_1+5x_2-8x_3=8\\ 4x_1+3x_2-9x_3=9\\ 2x_1+3x_2-5x_3=7\\ x_1+8x_2-7x_3=12 \end{cases}$$ From my elementary row operations, I get that it has no solution. (Row operations are to be ...