Math Problem Statement

Determine whether each equation has one solution, no solution, or infinite solutions: (a) 2(x - 1) + 6 = 4x - 22 (b) 6(2x + 1) - 2 = 12x + 4

Solution

Let's solve the equations step by step to determine whether each has one solution, no solution, or infinite solutions.


Part (a): 2(x1)+6=4x222(x - 1) + 6 = 4x - 22

  1. Expand the left-hand side: 2(x1)+6=2x2+6=2x+42(x - 1) + 6 = 2x - 2 + 6 = 2x + 4 So the equation becomes: 2x+4=4x222x + 4 = 4x - 22

  2. Move all xx-terms to one side and constants to the other: Subtract 2x2x from both sides: 4=2x224 = 2x - 22 Add 2222 to both sides: 26=2x26 = 2x

  3. Solve for xx: x=262=13x = \frac{26}{2} = 13

Conclusion: The equation has one solution, x=13x = 13.


Part (b): 6(2x+1)2=12x+46(2x + 1) - 2 = 12x + 4

  1. Expand the left-hand side: 6(2x+1)2=12x+62=12x+46(2x + 1) - 2 = 12x + 6 - 2 = 12x + 4 So the equation becomes: 12x+4=12x+412x + 4 = 12x + 4

  2. Simplify: Subtract 12x12x from both sides: 4=44 = 4

Conclusion: The equation is always true. Therefore, it has infinite solutions.


Final Answer:

  • (a) One solution: x=13x = 13
  • (b) Infinite solutions

Do you want details on any step or further explanations?


Here are 5 related questions to expand your understanding:

  1. What are the general steps to identify if an equation has one solution, no solution, or infinite solutions?
  2. How do you handle equations with fractions?
  3. What happens when variable terms cancel out entirely, leaving just numbers?
  4. How can you verify your solution by substitution?
  5. What are the key differences between equations and identities?

Tip: Always simplify both sides of an equation fully before deciding on the solution type.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Simplification of Equations

Formulas

Properties of equality (e.g., adding, subtracting, multiplying, and dividing both sides)

Theorems

Identity property (e.g., a = a)
No-solution property for inconsistent equations

Suitable Grade Level

Grade 8