https://file.aiquickdraw.com/mathbot/file/1726231626952y8nek040.jpg

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

### **1-14. Evaluate each expression if \( r = -3 \), \( s = 4 \), and \( t = 7 \):**
- **(a)** \( r^2 + \sqrt{s} \)
  \[
  (-3)^2 + \sqrt{4} = 9 + 2 = 11
  \]
- **(b)** \( \frac{t - r}{s} \)
  \[
  \frac{7 - (-3)}{4} = \frac{7 + 3}{4} = \frac{10}{4} = 2.5
  \]
- **(c)** \( 2s^2 + r - t \)
  \[
  2(4)^2 + (-3) - 7 = 2(16) - 3 - 7 = 32 - 3 - 7 = 22
  \]
- **(d)** \( 3(s - t)^2 \)
  \[
  3(4 - 7)^2 = 3(-3)^2 = 3(9) = 27
  \]

### **1-15. Diamond Problem Patterns**
- In these problems, the two numbers in the center must add up to give the number on the bottom and multiply to give the number on the top.
  
  **For the fourth diamond** (labeled with question marks):
  - You are given the two numbers (#), and you must find the numbers that satisfy both the sum and product rule.
  - Use the pattern from the first three diamonds:
    - First diamond: \( 5 + 7 = 12 \), \( 5 \times 7 = 35 \)
    - Second diamond: \( 2 + 3 = 5 \), \( 2 \times 3 = 6 \)
    - Third diamond: \( -4 + (-4) = -8 \), \( -4 \times (-4) = 16 \)

  To find the missing numbers, find two values that multiply to give the top number (#) and add to give the bottom number (#).

### **1-16. Solve for \( x \):**
- **(a)** \( x + 5 = 5 \)
  \[
  x = 5 - 5 = 0
  \]
- **(b)** \( 2x - 6 = 3x + 1 - x - 7 \)
  \[
  2x - 6 = 2x - 6
  \]
  This equation is always true, so all real numbers are solutions.
  
- **(c)** \( 3x + 1 = 43 \)
  \[
  3x = 43 - 1 = 42, \quad x = \frac{42}{3} = 14
  \]
- **(d)** \( 4x - 1 = 4x + 7 \)
  \[
  -1 \neq 7
  \]
  No solutions exist for this equation.

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Do you want more details on any of these, or do you have any questions?

Here are 5 related questions for further exploration:
1. What are other ways to check if the diamond problem pattern is valid for any given numbers?
2. How can you use the same pattern to solve for variables in more complex algebraic expressions?
3. Could you create a new diamond problem with different rules for summing and multiplying?
4. How would the solutions change if negative numbers were introduced in 1-16(c)?
5. In problem 1-14(c), how would the result change if \( t \) were negative?

**Tip:** Always check your work by plugging values back into the original equation to verify your solutions.

Evaluate algebraic expressions, identify patterns in diamond problems, and solve linear equations based on given information.

This guide walks you through solving algebraic expressions with exponents and square roots, recognizing patterns in diamond problems, and solving linear equations. Suitable for Grades 7-9, these problems enhance your understanding of core algebraic concepts.

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