Algebra 1 Questions & Answers

Recent questions in Algebra I
Zhyar Hoshyar Wali 2022-05-25

E.g. the equation is$u\left(x\right)=f\left(x\right){u}^{\prime }\left(x\right)$Is there a general form of solving such an equation?If not, is there a general form for $f\left(x\right)$ being a linear function?

Alani Conner 2022-05-23 Answered

For which value(s) of parameter m is there a solution for this system$\left\{\begin{array}{l}mx+y=m\\ mx+2y=1\\ 2x+my=m+1\end{array}$when does this system of equations have a solution?

Case Nixon 2022-05-23 Answered

Suppose that $x$, $y$ and $z$ are three integers (positive,negative or zero) such that we get the following relationships simultaneously1. $x+y=1-z$and2. ${x}^{3}+{y}^{3}=1-{z}^{2}$Find all such $x$, $y$ and $z$.

Nicholas Cruz 2022-05-23 Answered

Show that every irrational number in $\mathbb{R}$ is the limit of a sequence of rational numbers. Every rational number in $\mathbb{R}$ is the limit of a sequence of irrational numbers.

Cara Duke 2022-05-23 Answered

Proving $\sqrt{2}+\sqrt{3}$ is irrational

Isaiah Farrell 2022-05-23 Answered

Solve $\frac{1}{{x}^{2}-1}⩾\frac{1}{x+1}$

Brooke Ayala 2022-05-23 Answered

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\left({n}_{1},{n}_{2},{n}_{3},0\right)=1$$C\left({n}_{1},{n}_{2},{n}_{3},l\right)={n}_{1}C\left({n}_{1}-1,{n}_{2},{n}_{3},l-1\right)+{n}_{2}C\left({n}_{1}+1,{n}_{2}-1,{n}_{3},l-1\right)+{n}_{3}C\left({n}_{1},{n}_{2}+1,{n}_{3}-1,l-1\right)$The number of ways to choose the elements is then $C\left(0,0,d,l\right)$. Clearly there can be at most ${3}^{l}$ instances of the base case $C\left({n}_{1},{n}_{2},{n}_{3},0\right)=1$. 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?

Brooke Webb 2022-05-23 Answered

2022-05-22

why is 4/3 πr^3 a monomial

Brennen Fisher 2022-05-22 Answered