(x+y)+(x-2y)√2=2x-y+(x-y-1)√6

3 min read Jun 17, 2024

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

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) 
```
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.

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
```
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).

Solving for x in the second equation:
```
-x + y + 1 = 0
x = y + 1
```
Substituting x in the first equation:
```
(y + 1) - 2y = 0
-y + 1 = 0
y = 1
```
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.