https://file.aiquickdraw.com/mathbot/file/1729103864765p3m0u3ao.jpg

Let's work through the problems from the image:

### Problem 13:
Expression: \(3x^2 - y^2 + xy + 10\)

Given \(x = 2\) and \(y = -1\):
\[
3(2)^2 - (-1)^2 + (2)(-1) + 10
\]
\[
= 3(4) - 1 - 2 + 10
\]
\[
= 12 - 1 - 2 + 10 = 19
\]

**Solution for Problem 13**: 19

---

### Problem 14:
Expression: \(3x^2 - y^2 + xy + 10\)

Given \(x = 3\) and \(y = 2\):
\[
3(3)^2 - (2)^2 + (3)(2) + 10
\]
\[
= 3(9) - 4 + 6 + 10
\]
\[
= 27 - 4 + 6 + 10 = 39
\]

**Solution for Problem 14**: 39

---

### Problem 15: Solve the equations

1. \(y - 3 = 8\)
   \[
   y = 8 + 3 = 11
   \]

2. \(z - 7 = 20\)
   \[
   z = 20 + 7 = 27
   \]

3. \(\frac{p}{6} = 5\)
   \[
   p = 5 \times 6 = 30
   \]

4. \(\frac{m}{2} = 3\)
   \[
   m = 3 \times 2 = 6
   \]

5. \(9p = 64\)
   \[
   p = \frac{64}{9} \approx 7.11
   \]

6. \(x + 4 = 9\)
   \[
   x = 9 - 4 = 5
   \]

7. \(4x + 5 = 45\)
   \[
   4x = 45 - 5 = 40 \implies x = \frac{40}{4} = 10
   \]

8. \(3p + 8 = 20\)
   \[
   3p = 20 - 8 = 12 \implies p = \frac{12}{3} = 4
   \]

9. \(7 + m = 23\)
   \[
   m = 23 - 7 = 16
   \]

10. \(7b = 49\)
    \[
    b = \frac{49}{7} = 7
    \]

---

### Summary of Solutions:

- **Problem 13**: 19
- **Problem 14**: 39
- **Problem 15**:
  1. \(y = 11\)
  2. \(z = 27\)
  3. \(p = 30\)
  4. \(m = 6\)
  5. \(p \approx 7.11\)
  6. \(x = 5\)
  7. \(x = 10\)
  8. \(p = 4\)
  9. \(m = 16\)
  10. \(b = 7\)

Would you like more details or have any other questions?

### Related Questions:
1. How would you solve similar problems if the expressions were different?
2. What is the general approach for solving linear equations?
3. How does the order of operations affect the solution of equations?
4. What would happen if the variables had different values in Problems 13 and 14?
5. How can you check your solutions for accuracy?

### Tip:
Always follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) when evaluating expressions. This ensures accuracy in your calculations.

If x = 2 and y = -1, find the value of the following expression: 3x^2 - y^2 + xy + 10. If x = 3 and y = 2, find the value of the expression 3x^2 - y^2 + xy + 10. Solve the following equations: y - 3 = 8, z - 7 = 20, p/6 = 5, m/2 = 3, 9p = 64, x + 4 = 9, 4x + 5 = 45, 3p + 8 = 20, 7 + m = 23, 7b = 49.

Learn how to solve quadratic expressions like 3x^2 - y^2 + xy + 10 and linear equations step by step. These problems involve basic algebraic principles, suitable for students in Grades 6-8.

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