Understanding the Problem
We start with the equations given:
1. \(2x + y = \sqrt{8}\)
2. \(2x - y = \sqrt{8}\)
Both equations equal \(\sqrt{8}\).
Solving the Equations
We can solve these two equations simultaneously.
- Step 1: Add the Equations
- Adding the two equations:
\[
(2x + y) + (2x - y) = \sqrt{8} + \sqrt{8}
\]
This simplifies to:
\[
4x = 2\sqrt{8}
\]
- Step 2: Simplify for \(x\)
- Now, simplify \(\sqrt{8}\):
\[
\sqrt{8} = 2\sqrt{2}
\]
Thus:
\[
4x = 2 \times 2\sqrt{2} = 4\sqrt{2}
\]
Dividing both sides by 4 gives:
\[
x = \sqrt{2}
\]
Finding \(y\)
Next, we substitute \(x\) back into either equation to find \(y\). Using the first equation:
- Step 3: Substitute for \(y\)
\[
2(\sqrt{2}) + y = \sqrt{8}
\]
Simplifying gives:
\[
2\sqrt{2} + y = 2\sqrt{2}
\]
Thus:
\[
y = 2\sqrt{2} - 2\sqrt{2} = 0
\]
Final Values
The values of \(x\) and \(y\) are:
- \(x = \sqrt{2}\)
- \(y = 0\)
This solution satisfies both original equations, confirming its correctness.