🧮 algebra

1. **Stating the problems:** We have two quadratic equations to solve:

1. **State the problem:** We have the system of equations: $$2x + 3y = -8$$

1. **State the problem:** We are given a system of equations: $$y = 2 - 3x$$

1. **State the problem:** We are given the linear function $y = 2 - 3x$ and want to understand its properties. 2. **Formula and rules:** This is a linear equation in slope-intercep

1. **Problem statement:** Patrick spent $\frac{2}{5}$ of his salary on food, then $\frac{1}{3}$ of the remainder on electricity, and saved the rest. We need to find the fraction of

1. **State the problem:** Simplify the expression $$2\sqrt{27} - 2\sqrt{3} + \sqrt{12}\sqrt{75} + \sqrt{48} - 7\sqrt{3}$$. 2. **Recall important rules:**

1. **Stating the problem:** We are given several expressions of the form $\frac{a\pi + b}{\sqrt{2}}$ where $a$ and $b$ are constants. We want to understand or simplify these expres

1. **Problem:** Find the canonical form of the surface given by $$4x^2 + 9y^2 + 25z^2 = 900$$ and identify its type. 2. **Formula and rules:** The general form of an ellipsoid is $

1. The problem is to find the equation of a conic section given the focus at $(3,0)$, directrix $x=-3$, parameter $p=3$, latus rectum $4p=12$, and eccentricity $e=1$. 2. Since the

1. **State the problem:** We want to simplify and understand the function given by $$y=\frac{x^3}{x^2+1}$$. 2. **Formula and rules:** This is a rational function where the numerato

1. **State the problem:** We are given that $(x + 4)$ is a factor of the quadratic polynomial $x^2 - 3x + p$. We need to find the value of $p$. 2. **Recall the factor theorem:** If

1. **Stating the problem:** We are given an expression involving combinations (binomial coefficients) and algebraic terms:

1. **Stating the problem:** We need to find the odd one out among the following expressions: 1. $45^2$

1. **Problem 1:** Evaluate the expression using logarithm tables: $$\frac{3}{8.49 \times 6 \times 2.41} \times 3941$$

1. The problem asks if we can express $3^{n.9}$ as $3^2 \cdot 3^k$. 2. Recall the exponent rule: $a^m \cdot a^n = a^{m+n}$, which means when multiplying powers with the same base,

1. Simplify $y^2(-2y - 5y^2) + (-y^2 - y)$. Use distributive property: $y^2 \cdot (-2y) = -2y^3$, $y^2 \cdot (-5y^2) = -5y^4$.

1. **Problem 1:** Evaluate \( \frac{3}{8.496 \times 2.41} \times 3941 \) using logarithm tables. 2. **Problem 2:** Solve the equation \( \frac{3}{x-2} = \frac{4}{x-3} + 2 \).

1. **Stating the problem:** Mr. Nig Otiator has three types of cars: petrol, diesel, and electric. We know the number of cars that are not of each type: 24 are not petrol, 30 are n

1. **Stating the problem:** We want to verify or simplify the expression involving terms with powers of 3 and coefficients depending on $k$:

1. **Stating the problem:** We are given five equations and asked to find which one is the odd one out. 2. **Analyzing each equation:**

1. **Problem Statement:** Prove by induction that the sum of the series