Math Problem Statement

The image contains five mathematical problems. They include solving a system of equations, finding properties of points, differentiating a function, analyzing an expression, and evaluating integrals.

Solution

The image contains a set of mathematical problems. Let's break them down:

Task 1 (Зад. 1):

Solve the following system of linear equations:

3x_1 - x_2 + x_3 = 5 \\ x_1 + 2x_3 = 2 \\ x_1 - 2x_2 - x_3 = 0 \end{cases}$$ ### Task 2 (Зад. 2): Given points: - $$A(2, -2)$$ - $$B(-1, 0)$$ - $$C(3, -4)$$ Find some geometrical property (probably a line equation or something related). ### Task 3 (Зад. 3): The function given is: $$y = \frac{3x}{e^{2x + 1}}$$ Differentiate or analyze the behavior of the function. ### Task 4 (Зад. 4): The expression is: $$z = 2xy - x^2 + 3y^2 - 2x$$ Likely requiring partial derivatives or critical point analysis. ### Task 5 (Зад. 5): Three integrals: 1. $$\int \frac{dx}{x(1 + 2 \ln x)}$$ 2. $$\int (2 - x) e^{3x} \, dx$$ 3. $$\int_1^2 \frac{dx}{(4 - x)^3}$$ --- Which of these would you like to tackle first? Would you like details for one specific task?

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Coordinate Geometry
Differentiation
Partial Derivatives
Integrals

Formulas

System of Linear Equations
Derivative of exponential functions
Partial derivatives of multivariable functions
Standard integration techniques

Theorems

Cramer's Rule or Gaussian Elimination (for solving system of equations)
Fundamental Theorem of Calculus

Suitable Grade Level

University-level Calculus and Algebra

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