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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59
votes
15answers
9k views

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 ...
51
votes
5answers
4k views

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 ...
49
votes
8answers
6k views

Systems of linear equations: Why does no one plug back in?

When someone wants to solve a system of linear equations like $$\begin{cases} 2x+y=0 \\ 3x+y=4 \end{cases}\,,$$ they might use this logic: $$\begin{align} \begin{cases} 2x+y=0 \\ 3x+y=4 \end{cases}...
48
votes
7answers
5k views

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 ...
48
votes
1answer
991 views

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{...
44
votes
6answers
8k views

Find $xy+yz+zx$ given systems of three homogenous quadratic equations for $x, y, z$

This is a question from Math Olympiad. If $\{x,y,z\}\subset\Bbb{R}^+$ and if $$x^2 + xy + y^2 = 3 \\ y^2 + yz + z^2 = 1 \\ x^2 + xz + z^2 = 4$$ find the value of $xy+yz+zx$. I basically do not ...
41
votes
2answers
2k views

On Ramanujan's Question 359

In JIMS 4, p.78, Question 359 was asked by Ramanujan. (See The Problems Submitted by Ramanujan to the Journal of the Indian Mathematical Society, p. 9, by Bruce Berndt, et al.) If, $$\sin(x+y) = 2\...
35
votes
8answers
14k views

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 ...
34
votes
4answers
121k views

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 ...
32
votes
14answers
6k views

Are there any other methods to apply to solving simultaneous equations?

We are asked to solve for $x$ and $y$ in the following pair of simultaneous equations: $$\begin{align}3x+2y&=36 \tag1\\ 5x+4y&=64\tag2\end{align}$$ I can multiply $(1)$ by $2$, yielding $6x +...
32
votes
4answers
4k views

Super hard system of equations

Solve the system of equation for real numbers \begin{split} (a+b) &(c+d) &= 1 & \qquad (1)\\ (a^2+b^2)&(c^2+d^2) &= 9 & \qquad (2)\\ (a^3+b^3)&(c^3+d^3) &= 7 &...
30
votes
5answers
2k views

System of 4 tedious nonlinear equations: $ (a+k)(b+k)(c+k)(d+k) = $ constant for $1 \le k \le 4$

It is given that $$(a+1)(b+1)(c+1)(d+1)=15$$$$(a+2)(b+2)(c+2)(d+2)=45$$$$(a+3)(b+3)(c+3)(d+3)=133$$$$(a+4)(b+4)(c+4)(d+4)=339$$ How do I find the value of $(a+5)(b+5)(c+5)(d+5)$. I could think only of ...
28
votes
10answers
14k views

What makes a linear system of equations "unsolvable"?

I've been studying simple systems of equations, so I came up with this example off the top of my head: \begin{cases} x + y + z = 1 \\[4px] x + y + 2z = 3 \\[4px] x + y + 3z = -1 \end{cases} Combining ...
27
votes
4answers
9k views

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 ...
26
votes
4answers
2k views

How to prove this algebraic version of the sine law?

How to solve the following problem from Hall and Knight's Higher Algebra? Suppose that \begin{align} a&=zb+yc,\tag{1}\\ b&=xc+za,\tag{2}\\ c&=ya+xb.\tag{3} \end{align} Prove that $...
25
votes
3answers
32k views

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 ...
24
votes
5answers
2k views

Why does dividing both sides of this system of equations to each other yields infinite "incorrect solutions"?

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\...
23
votes
9answers
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 ...
23
votes
4answers
22k views

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, ...
22
votes
5answers
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 ...
20
votes
10answers
6k 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 ...
20
votes
6answers
778 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 ...
20
votes
2answers
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 ...
20
votes
2answers
875 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?
19
votes
2answers
2k views

Solving a system of equations with 3 variables in under a minute

...
19
votes
5answers
23k 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, ...
19
votes
1answer
32k 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(...
19
votes
0answers
417 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 ...
18
votes
8answers
6k 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 ...
17
votes
4answers
25k 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 ...
16
votes
2answers
4k views

Math Olympiad Algebra Question

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 ...
16
votes
1answer
212k views

What do trivial and non-trivial solution of homogeneous equations mean in matrices? [closed]

Suppose I have system of 3 equations $$a_1x+b_1y+c_1z=0$$ $$a_2x+b_2y+c_2z=0$$ $$a_3x+b_3y+c_3z=0$$ and cofficient matrix $A=\begin{equation} \begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 &...
15
votes
3answers
2k views

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 ...
15
votes
5answers
4k views

How to solve these $3$ equations for three unknowns $x$,$y$,$z$? [duplicate]

Question: Solve: $xy+x+y=23\tag{1}$ $yz+y+z=31\tag{2}$ $zx+z+x=47\tag{3}$ My attempt: By adding all we get $$\sum xy +2\sum x =101$$ Multiplying $(1)$ by $z$, $(2)$ by $x$, and $(3)$ by $y$ ...
15
votes
4answers
22k views

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 ...
14
votes
1answer
2k views

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 ...
14
votes
3answers
2k views

What is the most efficient way to find the inverse of large matrix?

Let $A$ be a large square $(n+1) \times (n+1)$ invertible matrix, where $n \approx 1000$. $$A = \begin{bmatrix} -1 & 0 & 0 &\cdots & 0 & a_0\\ 1 & -1 & 0 &\cdots & ...
14
votes
3answers
143k 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 \...
14
votes
5answers
375 views

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 ...
14
votes
4answers
565 views

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 ...
14
votes
2answers
23k views

Solve a linear system with more variables than equations

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: $$\...
14
votes
4answers
896 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)=...
13
votes
3answers
12k views

Determining a matrix from its characteristic polynomial

Let $A\in\mathcal{M}_{n}(K)$, where $K$ is a field. Then, we can obtain the characteristic polynomial of $A$ by simply taking $p(\lambda)=\det(A-\lambda I_n)$, which give us something like $$p(\lambda)...
13
votes
5answers
13k views

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 ...
13
votes
2answers
458 views

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$$ \left\{ \begin{array}{c} x_1+x_2=x_3^2 \\ x_2+x_3=x_4^2 \\ x_3+x_4=x_5^2 \\ x_4+x_5=x_1^2 \\ x_5+x_1=x_2^2 \end{array} \right. $$ ...
13
votes
3answers
545 views

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 ...
13
votes
1answer
407 views

Simultaneously vanishing quadratic forms

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 ...
12
votes
4answers
4k views

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 ...
12
votes
5answers
2k views

Find the value of $(\log_{a}b + 1)(\log_{b}c + 1)(\log_{c}a+1)$ if $\log_{b}a+\log_{c}b+\log_{a}c=13$ and $\log_{a}b+\log_{b}c+\log_{c}a=8$

Let $a,b$, and $c$ be positive real numbers such that $$\log_{a}b + \log_{b}c + \log_{c}a = 8$$ and $$\log_{b}a + \log_{c}b + \log_{a}c = 13.$$ What is the value of $$(\log_{a}b + 1)(\log_{b}c ...
12
votes
2answers
791 views

System of non-linear equations.

I have to find all triplets $(x,y,z)$ that satisfy: $$x^{2012} + y^{2012} + z^{2012} = 3\\x^{2013} + y^{2013} + z^{2013} = 3\\x^{2014} + y^{2014} + z^{2014} = 3$$ I've found the trivial solution $(1,...

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