https://file.aiquickdraw.com/mathbot/file/17268135116513c3iedy6.jpg

Let's go through the problems from the image one by one.

### Problem 15:
\[
\left\{
\begin{aligned}
x + y &= 1 \\
3x - 1 &= 9y
\end{aligned}
\right.
\]
We need to solve this system for \( y \).

### Problem 16:
Function: \( y = x^2 - 6x + 7 \) and the domain is \( 2 \leq x \leq 5 \). We are tasked with finding the range of \( y \).

### Problem 17:
Polynomial \( P(x) = x^3 + x^2 - 3x - 5 \) is divided by another polynomial \( Q(x) = x + 2 \). We need to find the remainder.

### Problem 18:
Sets: \( |A| = 8 \), \( |A \cap B| = 3 \), and \( |A \cup B| = 14 \). We need to find the cardinality \( |B| \).

### Problem 19:
Equation: \( x + y + z = 8 \), and we are asked to find how many natural solutions exist.

### Problem 20:
This is a geometric problem related to a cone. Given that the slant height is 8 and the radius of the base is 6, we need to calculate the angle of the sector that forms the lateral surface.

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Here are 5 questions that expand the current information:
1. How do you solve linear systems using substitution or elimination?
2. How do you find the range of a quadratic function?
3. What is polynomial division, and how do you calculate remainders?
4. How do you use set theory to solve problems involving intersections and unions of sets?
5. How do you count the number of natural solutions for equations like \(x + y + z = n\)?

**Tip:** In set theory, the principle of inclusion-exclusion is very useful for finding the size of unions and intersections of sets.

Solve the math problems shown in the image, covering algebraic systems, quadratic functions, polynomial division, set theory, and geometry.

This page covers multiple math problems including solving algebraic systems of equations, finding the range of quadratic functions, dividing polynomials, set theory applications, and geometric sector angle calculations. Detailed step-by-step solutions are provided for students in Grades 9-12.

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