Math Problem Statement

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

Solution

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

1-14. Evaluate each expression if r=3r = -3, s=4s = 4, and t=7t = 7:

  • (a) r2+sr^2 + \sqrt{s} (3)2+4=9+2=11(-3)^2 + \sqrt{4} = 9 + 2 = 11
  • (b) trs\frac{t - r}{s} 7(3)4=7+34=104=2.5\frac{7 - (-3)}{4} = \frac{7 + 3}{4} = \frac{10}{4} = 2.5
  • (c) 2s2+rt2s^2 + r - t 2(4)2+(3)7=2(16)37=3237=222(4)^2 + (-3) - 7 = 2(16) - 3 - 7 = 32 - 3 - 7 = 22
  • (d) 3(st)23(s - t)^2 3(47)2=3(3)2=3(9)=273(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=125 + 7 = 12, 5×7=355 \times 7 = 35
      • Second diamond: 2+3=52 + 3 = 5, 2×3=62 \times 3 = 6
      • Third diamond: 4+(4)=8-4 + (-4) = -8, 4×(4)=16-4 \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 xx:

  • (a) x+5=5x + 5 = 5 x=55=0x = 5 - 5 = 0

  • (b) 2x6=3x+1x72x - 6 = 3x + 1 - x - 7 2x6=2x62x - 6 = 2x - 6 This equation is always true, so all real numbers are solutions.

  • (c) 3x+1=433x + 1 = 43 3x=431=42,x=423=143x = 43 - 1 = 42, \quad x = \frac{42}{3} = 14

  • (d) 4x1=4x+74x - 1 = 4x + 7 17-1 \neq 7 No solutions exist for this equation.


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 tt were negative?

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

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Math Problem Analysis

Mathematical Concepts

Algebra
Arithmetic Operations
Linear Equations
Pattern Recognition

Formulas

r^2 + sqrt(s)
(t - r) / s
2s^2 + r - t
3(s - t)^2
Sum and Product Pattern in Diamond Problems

Theorems

Basic properties of exponents
Square root rules
Linear equation solving

Suitable Grade Level

Grades 7-9