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Algebra
Algebra I
Systems of equations

Recent questions in Systems of equations

Paiellorbkzp 2022-06-01

How to solve this sytem of equation:
${\begin{cases} (\sqrt{a^{2} + 4} + a) (\sqrt{b^{2} + 1} + b) = 1 \\ 27 a^{6} + 8 b = a^{3} + 2 \end{cases}$
I tried to analysis equation (1):
$\sqrt{a^{2} + 4} + a = \sqrt{b^{2} + 1} - b \Leftrightarrow a + b = \sqrt{b^{2} + 1} - \sqrt{a^{2} + 4}$
$\Leftrightarrow 4 a^{2} + 16 b^{2} + 20 a b = 9$
And then, I don't know how to solve this.

Davis Osborn 2022-06-01

How do I find the equilibria of this autonomous system of ODEs?
$x^{'} (t) = x (x^{3} - 2 y^{3})$
$y^{'} (t) = y (2 x^{3} - y^{3})$

Wotjake3jev8 2022-06-01

Solve simultaneously:
${\begin{array}{rcl} 6 (x + y) & = & 5 x y \\ 21 (y + z) & = & 10 y z \\ 14 (z + x) & = & 9 z x \end{array}$
Obviously one of the solutions is $(0, 0, 0)$ but I'm more interested in the other .
I actually solved this question , but using hard way.
I expanded each equation then got one variable in terms of other. For example after expanding first equation I get ,
$6 x + 6 y = 5 x y$
and then
$5 x y - 6 x = 6 y$
So,
$x (5 y - 6) = 6 y$
then,
$x = 6 y / (5 y - 6)$
Similarly I found the value of $x$ from the third equation ( in terms of $z$ ), equated both and got $z$ in terms of $y$ , substituted that in equation two and finally got the values. The solution is $(2, 3, 7)$ . You might have guessed that this is a very tedious method.

Timiavawsw9 2022-05-31

Is it immediately apparent that the solution to the system of equations,
$\begin{matrix} (1) & \begin{aligned} x_{1}^{2} & = x_{2} + 2 \\ x_{2}^{2} & = x_{3} + 2 \\ x_{3}^{2} & = x_{4} + 2 \\ ⋮ \\ x_{n}^{2} & = x_{1} + 2 \end{aligned} \end{matrix}$
can be given by the roots of unity? Specifically,
$\begin{matrix} (2a) & x = \frac{y_{k}^{2} + 1}{y_{k}} \end{matrix}$
where the $y_{k}$ are,
$\begin{matrix} (2b) & \begin{aligned} y_{k} & = \exp (\frac{2 π i k}{2^{n} - 1}), k = 0 \dots 2^{n - 1} - 1 \\ y_{k} & = \exp (\frac{2 π i k}{2^{n} + 1}), k = 1 \dots 2^{n - 1} \end{aligned} \end{matrix}$
Example. Let $n = 4$ . Then $(1)$ is equivalent to,
$\begin{matrix} (3) & x = (((x^{2} - 2)^{2} - 2)^{2} - 2)^{2} - 2 \end{matrix}$
Expanded out, $(3)$ is a $2^{4} = 16$ -deg polynomial and its 16 roots are given by $(2)$ where,
$y_{k} = \exp (\frac{2 π i k}{15}), k = 0 \dots 7$
$y_{k} = \exp (\frac{2 π i k}{17}), k = 1 \dots 8$
Ramanujan considered the system $(1)$ for $n = 3, 4$ in the general case and also as nested radicals. For $x = (((x^{2} - a)^{2} - a)^{2} - a)^{2} - a$ , see this related post. (Interestingly, $n = 5$ in the general case is no longer completely solvable in radicals.)
Question:
I observed $(2)$ empirically. How do we prove from first principles that this is indeed the solution?

Aiden Barry 2022-05-30

Solving a System of Equations with Cosine
${\begin{matrix} a + b = 77 ° \\ \cos (a) = \frac{y}{3.5} \\ \cos (a) = \frac{y + 1}{3.5 + x} \\ \cos (b) = \frac{y + 1}{3.5} \end{matrix}$

qtbabe9876a9 2022-05-30

calculate the value of $r$ and a from systems-of-equations given below?
$\frac{a (1 - r^{7})}{1 - r} = 86$
$\frac{a (1 - r^{10})}{1 - r} = - 682$

Rachel Villa 2022-05-30

Is there any nonempty, compact and invariant set in dynamical system generated by this system of equations?
$x^{'} = x + \sin (x y + 2) - 7$
My idea is to use this fact: Not empty omega limit set - because here we have also bounded functions and omega limit set is invariant. But it's hard to say anything about compactness.

res2bfitjq 2022-05-29

System of equations with n unknowns
How should explain solving a system of n unknowns we need n equations?

Mackenzie Rios 2022-05-28

Use Gaussian elimination to solve the system
$2 x - 3 y = - 5$
$3 x + y = 9$
Find the value of $x$ .

Anahi Jensen 2022-05-28

Visualise 3 simultaneous cubic equations
I have three equations of the form:
$\frac{i_{1}^{3}}{P_{1}} + i_{1} (Z_{1} + Z_{2}) + (i_{2} + i_{3}) Z_{2} - U_{1} = 0$
$\frac{i_{2}^{3}}{P_{2}} + i_{2} (Z_{1} + Z_{2}) + (i_{1} + i_{3}) Z_{2} - U_{2} = 0$
$\frac{i_{3}^{3}}{P_{3}} + i_{3} (Z_{1} + Z_{2}) + (i_{1} + i_{2}) Z_{2} - U_{3} = 0$
where $P_{1}, P_{2}, P_{3}, K, U_{1}, U_{2}, U_{3}, Z_{1}$ and $Z_{2}$ are all known constant complex numbers.
Can anyone recommend a way of visualising these three functions?

Nylah Burnett 2022-05-28

Get x and y in quadrat equations system
$2 x^{2} + y^{2} = 4$
$2 x y - 2 x = - 5$

Kenley Wagner 2022-05-28

If $a$ , $b$ , $c$ and $d$ satisfy the equations:
$a + 7 b + 3 c + 5 d = 0$
$8 a + 4 b + 6 c + 2 d = - 16$
$2 a + 6 b + 4 c + 8 d = 16$
$5 a + 3 b + 7 c + d = - 16$
then $(a + d) (b + c)$ equals $- 16$ .
why $(a + d) (b + c)$ equals $- 16$ ?

velitshh 2022-05-28

Solving systems of linear equations with an unknown 'a' using matrices and elementary row operations
Came across this one the other day... while I can narrow 'a' down I can't seem to find an exact/ optimised figure. For example 'a' cannot equal 1/3, 'a' must be less than 0.5...
Anyway, here's the problem. I've got a 3x3 matrix by a 3x1 which equates to a 3x1.
{0.6, 0.2, a},{0.4, 0.3, a},{0.0, 0.5, 1-2a}.{x, y, z} = {1360, 1260, 2000}

Alessandra Clarke 2022-05-28

Finding equilibrium for differential equations
$x^{'} = x^{2} + y^{2} - 5$
$y^{'} = x^{2} + 2 y^{2} - 9$
My goal is to find the equilibrium points.
$(\sqrt{5 - y^{2}}, \pm 2)$ , $(\pm \sqrt{\frac{9}{11}}, \sqrt{5 - x^{2}})$ , $(\sqrt{9 - 2 y^{2}}, \pm 2)$ and $(\pm \sqrt{\frac{11}{3}}, \frac{\sqrt{9 - x^{2}}}{2})$

Denisse Valdez 2022-05-27

System of equations:
$\dot{x} = x - y - x (x^{2} + y^{2}) + \frac{x y}{\sqrt{x^{2} + y^{2}}} \dot{y} = x + y - y (x^{2} + y^{2}) - \frac{x^{2}}{\sqrt{x^{2} + y^{2}}}$
How can I get this in polar coordinates? I know that $r^{2} = x^{2} + y^{2}$ , but how can I find $\dot{r}$ or $\dot{θ}$ ?

Alessandra Clarke 2022-05-27

Write the following system of equations in the form $A X = B$ , and calculate the solution using the equation $2 x - 3 y = - 1$ .
$2 x - 3 y = - 1$
$- 5 x + 5 y = 20$

Elisha Kelly 2022-05-27

system of equations solving for positive $a, b, c$
$(1 - a) (1 - b) (1 - c) = a b c$
$a + b + c = 1$
I found that $0 < a, b, c < 1$ and I try to solve it by try $(1 - a) = a$ , $(1 - b) = b$ and $(1 - c) = c$ and got that $a = 0.5, b = 0.5, c = 0.5$ but it contradicts the second equations.

Hailey Newton 2022-05-27

system of linear equations:
${\begin{cases} (1 + b) x + y + z = 1 \\ x + (1 + b) y + z = b \\ x + y + (1 + b) z = b^{2} \end{cases}$
And I would like to know if it has zero, one, or more solutions depanding on $b$ ( $b \in R$ ).
How do I do that?

Kaeden Woodard 2022-05-27

Solve the system of equations:
${\begin{cases} \frac{x^{2} + x y + y^{2}}{x^{2} + y^{2} + 1} = \frac{1}{x y} \\ {(\sqrt{3} + x y)}^{\log_{2} x} + \frac{x}{{(\sqrt{3} - x y)}^{\log_{2} y}} = 1 + \frac{x}{y} \end{cases}$

Jazmine Bruce 2022-05-27

Solve a system of two linear equations
If $2 x + 3 y = \frac{1}{10}$ and $3 x + 2 y = \frac{1}{8}$ , then how to calculate here $x$ and $y$ ?

In simple terms, systems of equations represent a special set of simultaneous equations where the equation system is used as a finite element. The trick here is to find common solutions, which is exactly what systems of equations solver must achieve. If this does not sound clear to you, take a look at some systems of equations answers below and see those with explanations. The solution will always come in three variables (namely, x, y, and z), which will represent your ordered triple. See systems of equations solutions for more examples of how it works in practice.

Algebra I

Strengthen Your Systems of Equations Skills