(x+y)+(x-2y)√2=2x-y+(x-y-1)√6

3 min read Jun 17, 2024
(x+y)+(x-2y)√2=2x-y+(x-y-1)√6

Solving the Equation (x+y)+(x-2y)√2=2x-y+(x-y-1)√6

This equation involves both rational and irrational terms, making it a bit tricky to solve. Here's a breakdown of how to approach it:

Isolating the Irrational Terms

  1. Rearrange the equation: Start by moving all terms with √2 to one side and all terms with √6 to the other.

    (x-2y)√2 - (x-y-1)√6 = 2x-y - (x+y) 
    
  2. Simplify: Combine like terms on both sides.

    (x-2y)√2 - (x-y-1)√6 = x - 2y
    

Comparing Coefficients

Now, we need to consider that the equation must hold true regardless of the value of √2 and √6. This means that the coefficients of √2 and √6 on both sides must be equal.

  1. Coefficients of √2: The coefficient of √2 on the left side is (x-2y). There is no √2 term on the right side, so its coefficient is 0. Therefore:

    x - 2y = 0
    
  2. Coefficients of √6: The coefficient of √6 on the left side is -(x-y-1). There is no √6 term on the right side, so its coefficient is 0. Therefore:

    -(x-y-1) = 0
    

Solving the System of Equations

We now have two equations:

  • x - 2y = 0
  • -(x-y-1) = 0

This is a system of linear equations that can be solved using various methods (substitution, elimination).

  1. Solving for x in the second equation:

    -x + y + 1 = 0
    x = y + 1
    
  2. Substituting x in the first equation:

    (y + 1) - 2y = 0
    -y + 1 = 0
    y = 1
    
  3. Substituting y back into the equation for x:

    x = 1 + 1
    x = 2
    

Solution

Therefore, the solution to the equation is: x = 2 and y = 1.

Note: It's always a good idea to check your solution by plugging the values of x and y back into the original equation to ensure it holds true.

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