Math Problem Statement

Solve the equations provided in the image for the variables, first combining like terms or expanding brackets, and provide the solution sets.

Solution

I see the mathematical exercises from the uploaded image. Let’s break down and explain the solutions for both exercises.

Exercise 5:

You are asked to combine like terms and solve the equations, giving the solution set.

a) 2x+5+3+x+7x=882x + 5 + 3 + x + 7x = 88

  1. Combine all terms with xx: 2x+x+7x=10x2x + x + 7x = 10x
  2. Combine constants: 5+3=85 + 3 = 8 Equation simplifies to: 10x+8=8810x + 8 = 88
  3. Subtract 8 from both sides: 10x=8010x = 80
  4. Divide by 10: x=8x = 8 Solution set: L={8}L = \{8\}

b) 7x+4x+9+12+5=377x + 4x + 9 + 12 + 5 = 37

  1. Combine all terms with xx: 7x+4x=11x7x + 4x = 11x
  2. Combine constants: 9+12+5=269 + 12 + 5 = 26 Equation simplifies to: 11x+26=3711x + 26 = 37
  3. Subtract 26 from both sides: 11x=1111x = 11
  4. Divide by 11: x=1x = 1 Solution set: L={1}L = \{1\}

Exercise 6:

You are tasked with solving equations by expanding the brackets and finding xx.

a) 3(x+5)=303(x + 5) = 30

  1. Distribute 33: 3x+15=303x + 15 = 30
  2. Subtract 15 from both sides: 3x=153x = 15
  3. Divide by 3: x=5x = 5

b) 6(x+4)=606(x + 4) = 60

  1. Distribute 66: 6x+24=606x + 24 = 60
  2. Subtract 24 from both sides: 6x=366x = 36
  3. Divide by 6: x=6x = 6

c) 3(x5)=303(x - 5) = 30

  1. Distribute 33: 3x15=303x - 15 = 30
  2. Add 15 to both sides: 3x=453x = 45
  3. Divide by 3: x=15x = 15

d) 4(x5)=84(x - 5) = 8

  1. Distribute 44: 4x20=84x - 20 = 8
  2. Add 20 to both sides: 4x=284x = 28
  3. Divide by 4: x=7x = 7

Would you like any of these steps detailed further? Here are five questions you might want to explore:

  1. How do we solve equations with fractions or decimals?
  2. Can you explain the distributive property further?
  3. What happens when there are variables on both sides of the equation?
  4. How do we solve inequalities using similar steps?
  5. Can we graph the solutions to these equations?

Tip: Always check your solution by substituting xx back into the original equation to ensure it satisfies it!

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Math Problem Analysis

Mathematical Concepts

Algebra
Equation Solving
Distributive Property

Formulas

ax + b = c
a(b + c) = ab + ac

Theorems

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Suitable Grade Level

Grades 7-8