🧮 algebra

1. **Stating the problem:** We need to find the shape of the region defined by the system of inequalities:

1. We are given the system of equations: $$x + 2y = 6$$

1. Diberikan persamaan: $$5^{x-2y+1} = 25^{x-2y}$$

1. Diberikan persamaan: $$5^{x-2y+1} = 25^{x-2y}$$

1. Diketahui fungsi kuadrat $f(x) = ax^2 + bx + c$ yang melalui titik $(0, 3)$ dan mencapai minimum di titik $(-2, 1)$. Kita diminta mencari nilai $a - b + c$. 2. Karena fungsi kua

1. Diberikan persamaan kuadrat $$x^2 - 5x - 1 = 0$$ dengan akar-akar $$x_1$$ dan $$x_2$$. Kita diminta mencari nilai $$x_1^2 - 4x_1 + x_2$$. 2. Dari sifat akar-akar persamaan kuadr

1. **State the problem:** Given two functions $f(x) = x^2$ and $g(x) = 2x + 1$, find the following combined functions: - $(f + g)(x)$

1. Problem: Find $(f+g)(1)$ where $f(x) = \frac{3x+1}{x+1}$ and $g(x) = \frac{x^{-5}}{x}$.\n\n2. Formula: $(f+g)(x) = f(x) + g(x)$. We evaluate each function at $x=1$ and then add.

1. **State the problem:** Evaluate $\ln\left(e^{\sqrt{3}}\right)$.\n\n2. **Recall the logarithm rule:** For any positive number $a$ and exponent $b$, $\ln\left(e^b\right) = b$ beca

1. **State the problem:** We need to solve the equation $$\frac{2x+4}{x-1} = 3$$ for $x$. 2. **Formula and rules:** To solve rational equations, multiply both sides by the denomina

1. **Problem:** Solve the equation $$\left(\frac{16}{18}\right)^{x-1} = \left(\frac{6}{9}\right)^{\frac{x}{2}}$$. 2. **Step 1: Simplify the bases.**

1. **State the problem:** Simplify the fraction $\frac{231}{1001}$ to its lowest terms and find the least common denominator (LCD) for the given options. 2. **Prime factorization:*

1. **State the problem:** Reduce the fraction $\frac{231}{1001}$ to its lowest terms. 2. **Formula and rules:** To reduce a fraction, divide numerator and denominator by their grea

1. **State the problem:** Simplify the expression $$\frac{18a - 5ab}{6a^2}$$. 2. **Identify the terms:** The numerator is $18a - 5ab$ and the denominator is $6a^2$.

1. المشكلة: حل المتتاليات وتعريفها. 2. المتتالية هي ترتيب من الأعداد يُكتب فيه كل عدد في مكان معين يسمى حد المتتالية.

1. **مشكلة:** أكمل التعريفات والعبارات المتعلقة بالدوال والمصفوفات والاحتمالات. 2. **الدالة الخطية تعرف بما يلي:**

1. **Problem:** Given matrices \(P = \begin{pmatrix}3 & -4 \\ 5 & 1\end{pmatrix}\) and \(Q = \begin{pmatrix}2 \\ 8\end{pmatrix}\), find \(P^{-1}\). 2. **Formula for inverse of a 2x

1. **State the problem:** We need to determine which number line correctly displays the points $\{-1.70, \frac{1}{2}, -1.25, -\frac{3}{5}\}$ in order from left to right. 2. **Conve

1. **Problem statement:** A batsman has an average of 37 runs after 17 innings. We want to find how many runs he should score in the next innings so that his average increases by 6

1. **Problem statement:** A batsman has an average of 37 runs after 17 innings. We want to find how many runs he should score in the next innings so that his average increases by 6

1. The problem asks to determine the truth value of several algebraic and geometric statements. 2. Statement 1: For any integers $a$, $b$, and $c$, $a(b + c) = ab + ac$.