Tabansi
2020-12-17
Answered

Write the vector form of the general solution of the given system of linear equations.

$3{x}_{1}+{x}_{2}-{x}_{3}+{x}_{4}=0$

$2{x}_{1}+2{x}_{2}+4{x}_{3}-6{x}_{4}=0$

$2{x}_{1}+{x}_{2}+3{x}_{3}-{x}_{4}=0$

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Answered 2020-12-18
Author has **108** answers

Write the augmented matrix of the coefficients and constants

Transform the matrix in its reduced row echelon form.

Determine the general solution

Rewrite the solution in vector form

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Find the linear approximation of the function

Use L(x) to approximate the numbers

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Sugar maples are 8.4 meters tall. Telephone company plans to remove the top third of the trees.

Find the height of sugar maples after they are shortened.

Find the height of sugar maples after they are shortened.

asked 2020-11-06

Let G be a group. Show that $G\ne qH\cup KG=H\cup K$ for any two subgroups $H\le GH\le G{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}K\le GK\le G$ .

asked 2022-01-04

If ${a}_{1},{a}_{2},{a}_{3},\dots {\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}{b}_{1},{b}_{2},{b}_{3},\dots$ are arithmetic sequences, show that ${a}_{1}+{b}_{1},{a}_{2}+{b}_{2},{a}_{3}+{b}_{3},\dots$ is also an arithmetic sequence.

asked 2022-05-15

Three-variable system of simultaneous equations

$x+y+z=4$

${x}^{2}+{y}^{2}+{z}^{2}=4$

${x}^{3}+{y}^{3}+{z}^{3}=4$

Any ideas on how to solve for $(x,y,z)$ satisfying the three simultaneous equations, provided there can be both real and complex solutions?

$x+y+z=4$

${x}^{2}+{y}^{2}+{z}^{2}=4$

${x}^{3}+{y}^{3}+{z}^{3}=4$

Any ideas on how to solve for $(x,y,z)$ satisfying the three simultaneous equations, provided there can be both real and complex solutions?

asked 2021-10-16

Simplify each expression. Express final results without using zero or negative integers as exponents.

asked 2022-05-23

I want to know how many ways there are to choose $l$ elements in order from a set with $d$ elements, allowing repetition, such that no element appears more than 3 times. I've thought of the following recursive function to describe this:

$C({n}_{1},{n}_{2},{n}_{3},0)=1$

$C({n}_{1},{n}_{2},{n}_{3},l)={n}_{1}C({n}_{1}-1,{n}_{2},{n}_{3},l-1)+{n}_{2}C({n}_{1}+1,{n}_{2}-1,{n}_{3},l-1)+{n}_{3}C({n}_{1},{n}_{2}+1,{n}_{3}-1,l-1)$

The number of ways to choose the elements is then $C(0,0,d,l)$. Clearly there can be at most ${3}^{l}$ instances of the base case $C({n}_{1},{n}_{2},{n}_{3},0)=1$. Additionally, if ${n}_{i}=0$, that term will not appear in the expansion since zero times anything is zero.

It isn't too hard to evaluate this function by hand for very small l or by computer for small l, but I would like to find an explicit form. However, while I know how to turn recurrence relations with only one variable into explicit form by expressing them as a system of linear equations (on homogeneous coordinates if a constant term is involved) in matrix form, I don't know how a four variable equation such as this can be represented explicitly. There's probably a simple combinatorical formulation I'm overlooking. How can this function be expressed explicitly?

$C({n}_{1},{n}_{2},{n}_{3},0)=1$

$C({n}_{1},{n}_{2},{n}_{3},l)={n}_{1}C({n}_{1}-1,{n}_{2},{n}_{3},l-1)+{n}_{2}C({n}_{1}+1,{n}_{2}-1,{n}_{3},l-1)+{n}_{3}C({n}_{1},{n}_{2}+1,{n}_{3}-1,l-1)$

The number of ways to choose the elements is then $C(0,0,d,l)$. Clearly there can be at most ${3}^{l}$ instances of the base case $C({n}_{1},{n}_{2},{n}_{3},0)=1$. Additionally, if ${n}_{i}=0$, that term will not appear in the expansion since zero times anything is zero.

It isn't too hard to evaluate this function by hand for very small l or by computer for small l, but I would like to find an explicit form. However, while I know how to turn recurrence relations with only one variable into explicit form by expressing them as a system of linear equations (on homogeneous coordinates if a constant term is involved) in matrix form, I don't know how a four variable equation such as this can be represented explicitly. There's probably a simple combinatorical formulation I'm overlooking. How can this function be expressed explicitly?