🧮 algebra

1. **State the problem:** Simplify or analyze the expression $2x + y + 1$. 2. **Understand the expression:** This is a linear expression in two variables $x$ and $y$ with a constan

1. The problem is to find the limit as $n$ approaches infinity of the sequence $$1 + 2 + 2^2 + \dots + 2^{n-1} - 3 \cdot 2^{n+1}.$$\n\n2. This is a sum of a geometric series minus

1. **Тодорхойлолт:** $\lim_{n \to \infty} 1 + 2 + 2^2 + \dots + 2^{n-1} - 3 \cdot 2^{n+1}$ дарааллын хязгаарыг олох. 2. **Формул ба дүрэм:** Геометр прогрессын нийлбэрийн томъёо аш

1. **State the problem:** We know the car's value in 2021 is 16000 and it depreciates by 12.5% each year. We want to find its value in 2016. 2. **Formula used:** The value after $t

1. Stating the problem: Solve the system of linear equations using Gaussian elimination: $$\begin{cases} x_1 + 4x_2 + 2x_3 = 20 \\ -x_1 - 5x_2 + 3x_3 = -24 \\ 3x_1 - 3x_2 + 4x_3 =

1. **State the problem:** Solve the system of linear equations using Gauss elimination: $$\begin{cases}-2b + 10c = 4 \\ 3a + 12b - 6c = -9 \\ 9a + 34b - 7c = -19 \end{cases}$$

1. The problem is to find the system of linear equations corresponding to the given augmented matrix: $$\begin{bmatrix}4 & 0 & 0 \\ 7 & -4 & 0 \\ 0 & 3 & 3\end{bmatrix}$$

1. **State the problem:** We want to find coefficients $a$, $b$, and $c$ such that the system $$

1. **State the problem:** Determine the degree of homogeneity of the function $$f(x,y) = x \ln(x) - y \ln(y)$$. 2. **Recall the definition:** A function $$f(x,y)$$ is homogeneous o

1. **State the problem:** Simplify the expression $6a \times \frac{5b^2}{3a^3}$. 2. **Write the expression:**

1. **State the problem:** Solve the equation $$12(2k + 11) = 12(2k + 12)$$ for $k$. 2. **Write the equation:** $$12(2k + 11) = 12(2k + 12)$$

1. **Problem statement:** Solve the system of linear equations $$\begin{cases} x_1 + x_2 + 2x_3 = -1 \\ 2x_1 - x_2 + 2x_3 = -4 \\ 4x_1 + x_2 + 4x_3 = -2 \end{cases}$$

1. Muammo: Chiziqli tenglamalar sistemasini a) Gauss usuli yordamida, b) Kramer usuli yordamida yeching. 2. Tenglamalar sistemasini yozamiz: $$\begin{cases} 2x + y = 3 \\ x - y = 1

1. **State the problem:** A hill on a dirt bike course is 5 feet tall. The valley below the hill is as deep as the hill is tall. We need to find the integer representing the locati

1. The problem states that Chad is planting a plant that is 4 inches tall and wants the hole to be as deep as the plant is tall. 2. We need to find the integer that represents the

1. Problem statement: The goal is to use $y$ instead of $f(x)$ to name a function and to show how to rewrite, evaluate, and solve for variables. 2. Formula and rules: A function wr

1. **State the problem:** Graph the inequality $$-x + 3y > -12$$. 2. **Rewrite the inequality in slope-intercept form:**

1. **State the problem:** Find the inverse function $h^{-1}(x)$ of the one-to-one function $h(x) = \frac{5x - 3}{6x + 1}$ and determine the domain and range of $h^{-1}$. 2. **Formu

1. Muammo: 4x4 matritsaning determinantini hisoblash kerak. Matritsa:

1. **State the problem:** Solve the equation $$\sqrt{x}2x + 63 = x$$ for $x$. 2. **Rewrite the equation:** The equation is $$2x\sqrt{x} + 63 = x$$.

1. **State the problem:** Factor the quadratic expression $9a^2 + 9a + 36$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for factors of $a \times c