🧮 algebra

1. **State the problems:** We need to simplify two expressions:

1. **State the problem:** Solve the system of linear equations: $$5y + x = 44$$

1. **State the problem:** Solve the system of equations using the augmented matrix method: $$\begin{cases} x + 2y - z = 3 \\ x + 3y + z = 5 \\ 3x + 8y + 4z = 17 \end{cases}$$

1. **State the problem:** Find the horizontal and vertical asymptotes of the function $$f(x) = \frac{2x^2 - x + 1}{x - 2}$$. 2. **Recall definitions:**

1. Problema este să găsim suma $x + y + z$ dat fiind că $x \cdot y + x \cdot z + y \cdot z = 3 x \cdot y \cdot z$. 2. Formula dată este o ecuație simetrică în $x, y, z$. O metodă c

1. **Evaluate powers of i and negative powers:** Recall that $i = \sqrt{-1}$ and powers of $i$ cycle every 4: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, then repeats.

1. **State the problem:** Simplify the following expressions involving powers of the imaginary unit $i$: i) $i^9$

1. The problem is to find the cube root of 64. 2. The cube root of a number $x$ is a number $y$ such that $y^3 = x$.

1. Muammo: Geometrik progressiyaning maxraji $q=3$, dastlabki to'rta hadlari yig'indisi $S_4=80$. To'rtinchi hadni toping. Formulalar:

1. Masala: Geometrik progressiyada uchinchi va yettinchi hadlarining ko’paytmasi 144 ga teng. Beshinchi hadini toping. Formulalar: Geometrik progressiyaning $n$-chi hadi $b_n = b_1

1. Muammo: Geometrik progressiyaning dastlabki 4 ta hadiga mos ravishda 1; 1; 4 va 13 sonlarini qo’shsak, ular arifmetik progressiyani tashkil etadi. Geometrik progressiyaning maxr

1. Masala: Geometrik progressiyaning dastlabki 6 ta hadi $b_1, b_2, b_3, b_4, b_5, b_6=486$ berilgan. $b_2+b_3+b_4+b_5$ ni toping. Formulalar: Geometrik progressiya hadlari $b_n =

1. **Problem Statement:** Find the power of a product, i.e., simplify the expression $$(ab)^n$$ where $a$ and $b$ are any numbers and $n$ is a positive integer. 2. **Formula Used:*

1. **Problem Statement:** Find the total number $N$ of 5-digit numbers divisible by 4 formed using digits 1, 2, 3, 4, 5, and 6 without repetition. 2. **Key Rule for Divisibility by

1. **Problem Statement:** Answer all parts of Exercise 1.3 related to linear inequalities and systems. 2. **(1)(a) Boundary line of $ax + b \geq 0$:**

1. The problem asks to find $f(16)$ given the function $f(x) = x^{1/4}$. 2. The function $f(x) = x^{1/4}$ means the fourth root of $x$, or equivalently $f(x) = \sqrt[4]{x}$.

1. **Stating the problem:** We want to prove the identity $$\left(\frac{a}{b}\right)^{-\frac{m}{n}} = \left(\frac{b}{a}\right)^{\frac{m}{n}}$$ where $a$, $b$, $m$, and $n$ are real

1. **State the problem:** Ris uses three-fifths of a bag of flour to make pancakes, and after that, 600g of flour is left. We need to find out how much flour she used. 2. **Set up

1. **State the problem:** Solve the linear equation $2x + 2y = 36$ for $y$ in terms of $x$. 2. **Formula and rules:** To solve for $y$, isolate $y$ on one side of the equation. Thi

1. Problem (a): Simplify $2 \frac{1}{2} \div \left(\frac{3}{4} - \frac{1}{8}\right)$. 2. Convert mixed number to improper fraction: $2 \frac{1}{2} = \frac{5}{2}$.

1. **Problem Statement:** A company buys cars for 28000 each and depreciates them straight-line over 7 years, losing 4000 in value each year. We need to: