# 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 can I prove that 3 planes are arranged in a triangle-like shape without calculating their intersection lines?

The problem So recently in school, we should do a task somewhat like this (roughly translated): Assign a system of linear equations to each drawing Then, there were some systems of three linear ...
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### Hahn-Banach From Systems of Linear Equations

In this paper1 on the history of functional analysis, the author mentions the following example of an infinite system of linear equations in an infinite number of variables $c_i = A_{ij} x_j$: \begin{...
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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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### Balance chemical equations without trial and error?

In my AP chemistry class, I often have to balance chemical equations like the following: $$\mathrm{Al} + \text O_2 \to \mathrm{Al}_2 \mathrm O_3$$ The goal is to make both side of the arrow have ...
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### Fast algorithm for solving system of linear equations

I have a system of $N$ linear equations, $Ax=b$, in $N$ unknowns (where $N$ is large). If I am interested in the solution for only one of the unknowns, what are the best approaches? For example, ...
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This might be basic, but I'm really bad at basic math. I'm trying to solve the following system of equations: $$\sqrt{x^2+y^2}\cdot \left(x-5\right)=6x+y \tag{1},$$$$\\\sqrt{x^2+y^2}\cdot \left(y-1\... • 713 24 votes 1 answer 277k views ### What do trivial and non-trivial solution of homogeneous equations mean in matrices? [closed] Suppose I have system of 3 equations$$a_1x+b_1y+c_1z=0a_2x+b_2y+c_2z=0a_3x+b_3y+c_3z=0$$and cofficient matrix A= \begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 &... • 2,156 23 votes 5 answers 1k views ### Solving a peculiar system of equations I have the following system of equations where the m's are known but a, b, c, x, y, z are unknown. How does one go about solving this system? All the usual linear algebra tricks I know don't apply ... • 2,015 22 votes 9 answers 4k views ### Finding three unknowns from three equations. Solvable? If so, how? I have the following three equations: \begin{cases} v_{1f}\cos(37^\circ)+v_{2f}\cos(\theta) & = 3.5 \times 10^5 \\ v_{1f}\sin(37^\circ)-v_{2f}\sin(\theta) & = 0\\ v_{1f}^2+v_{2f}^2 & =(3.5 ... • 719 22 votes 4 answers 31k views ### Proof that any linear system cannot have exactly 2 solutions. How would you go about proving that for any system of linear equations (whether all are homogenous or not) can only have either (if this is true): One solution Infinitely many solutions No solutions ... • 3,098 22 votes 1 answer 38k views ### Polar coordinates differential equation I have the following ODE:$$\dot x=-y(x^2+y^2), \dot y=x(x^2+y^2)I want to sketch the phase portrait (manually) and I want to find the flow \phi_t, the orbit O(x_0) and the limit set \omega(... • 647 22 votes 2 answers 985 views ### Does this system of simultaneous Pell-like equations have any non-trivial positive integer solutions? Let a,b,c be positive integers satisfying \begin{align} 2a^2-1 &= b^2, \\ 2a^2+1 &= 3c^2. \end{align} The trivial solution is (a,b,c)=(1,1,1). Are there others? • 7,969 22 votes 1 answer 561 views ### Four squares such that the difference of any two is a square? I. This post asks to find 4 integers a,b,c,d such that the difference between any two is a square. As mentioned by my answer, it is equivalent to finding 3 squares such that the difference of ... 21 votes 10 answers 7k views ### System of nonlinear equations that leads to cubic equation The system of equations are:\begin{align}2x + 3y &= 6 + 5x\\x^2 - 2y^2 - (3x/4y) + 6xy &= 60\end{align}$$I can solve it through substitution but it is an arduous process to reach this ... • 489 20 votes 5 answers 31k views ### Do row operations change the column space of a matrix? I know that (i) row operations do not change the row space (ii) column operations do not change the column space and (iii) row rank = column rank (but this is sort of unrelated, I think). But, ... 20 votes 2 answers 1k views ### Is the AM-GM inequality the only obstruction for getting a specific sum and product? This might be silly, but here it goes. Let P,S>0 be positive real numbers that satisfy \frac{S}{n} \ge \sqrt[n]{P}. Does there exist a sequence of positive real numbers a_1,\dots,a_n such ... • 25.4k 20 votes 5 answers 600 views ### If \sin x+\sin y+\sin z=2, \cos x+\cos y+\cos z=11/5, \tan x+\tan y+\tan z=17/6, x,y,z\in\mathbb{R}, find \sin(x+y+z) without a calculator Given$$\begin{align} \sin x+\sin y+\sin z &=2 \\[4pt] \cos x+\cos y+\cos z &=\frac{11}{5} \\[4pt] \tan x+\tan y+\tan z &=\frac{17}{6} \end{align}$$where x,y,z\in\mathbb{R}. Find the ... • 25.9k 19 votes 2 answers 3k views ### Solving a system of equations with 3 variables in under a minute ... • 387 19 votes 8 answers 7k views ### Kid's homework: 4 equations 5 unknowns? Going crazy! I'm new here, and I'm hoping someone can help out. My 10 year old son has been set a maths problem, which I can't solve. I've got a PhD in neuroscience and do a fair amount of matlab stuff (data ... • 207 19 votes 6 answers 925 views ### Solve the system of equations for \sqrt{xy}$$x + y\sqrt{x} = \frac{95}{8}y + x\sqrt{y} = \frac{93}{8}x, y \in \mathbb{R}$$I can't solve this system of equations I got asked in a group. I added and substracted them to find the ... • 657 17 votes 3 answers 182k views ### Set of Linear equation has no solution or unique solution or infinite solution? For the system$$ \left\{ \begin{array}{rcrcrcr} x &+ &3y &- &z &= &-4 \\ 4x &- &y &+ &2z &= &3 \\ 2x &- &y &- &3z &= &1 \...
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If, $$ax + by = 7$$ $$ax^2 + by^2 = 49$$ $$ax^3 + by^3 = 133$$ $$ax^4 + by^4 = 406$$ Then find the value of$-$ $$2014(x+y-xy) - 100(a+b)$$ I came across this question in a Math Olympiad ...
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### Proving that two systems of linear equations are equivalent if they have the same solutions

I've just begun to work learn Linear Algebra on my own through Hoffman and Kunze's book and the first problem set already has a question that I can't solve: Prove that if two homogeneous systems of ...
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### Easiest way to solve this system of equations

I have these two equations: $$x=\frac{ab(1+k)}{b+ka}\\ y=\frac{ab(1+k)}{a+kb}$$ where $a,b$ are constants and $k$ is a parameter to be eliminated. A relation between $x,y$ is to be found. What is ...
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### Solve the system $x \lfloor y \rfloor = 7$ and $y \lfloor x \rfloor = 8$.

Solve the following system for $x,y \in \mathbb{R}$: \begin{align} x \lfloor y \rfloor & = 7, \\ y \lfloor x \rfloor & = 8. \end{align} It could be reducing to one variable, but it is not ...
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### Solving an equality with 3 equations, and 3 variables

Following is the question asked in a recent aptitude exam: Given that : $$a+b+ab=10\\ b+c+bc=20 \\ c+a+ac=30$$ What is the value of $a+b+c+abc$ ? I can solve it by finding the individual values ...
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### Solve $\begin{cases}x^2+y^4=20\\x^4+y^2=20\end{cases}$

Solve $$\begin{cases}x^2+y^4=20\\x^4+y^2=20\end{cases}.$$ I was thinking about letting $x^2=u,y^2=v.$ Then we will have $$\begin{cases}u+v^2=20\Rightarrow u=20-v^2\\u^2+v=20\end{cases}.$$ If we ...
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Suppose that, after a series of elementary row operations the augmented matrix of a linear system with variables $x_1$, $x_2$, $x_3$, $x_4$ is transformed into reduced row echelon form as follows: $$\... • 926 14 votes 5 answers 5k views ### How to solve an exponential and logarithmic system of equations?$$ \left\{\begin{array}{c} e^{2x} + e^y = 800 \\ 3\ln(x) + \ln(y) = 5 \end{array}\right.$$I understand how to solve system of equations, logarithmic rules, and the fact that \ln(e^x) = e^{\ln(x)} ... • 477 14 votes 4 answers 939 views ### How find the value of the x+y Question: let x,y\in \Bbb R , and such$$\begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases}$$Find the x+y This problem is from china some BBS My idea: since$$(3x^3+4y^3)(4x^4+3y^4)=...
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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 ...
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### What am I doing wrong solving this system of equations?

$$\begin{cases} 2x_1+5x_2-8x_3=8\\ 4x_1+3x_2-9x_3=9\\ 2x_1+3x_2-5x_3=7\\ x_1+8x_2-7x_3=12 \end{cases}$$ From my elementary row operations, I get that it has no solution. (Row operations are to be ...
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### Finding two numbers given their sum and their product

Which two numbers when added together yield $16$, and when multiplied together yield $55$. I know the $x$ and $y$ are $5$ and $11$ but I wanted to see if I could algebraically solve it, and found I ...
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### Prove that the system $A^T A x = A^T b$ always has a solution
Prove that the system $$A^T A x = A^T b$$ always has a solution. The matrices and vectors are all real. The matrix $A$ is $m \times n$. I think it makes sense intuitively but I can't prove it ...