Sure, let's solve the given equation step by step:
The given equation is:
[tex]\[ x - 3 + \sqrt{y + 2} = -1 + \sqrt{3} \][/tex]
### Step 1: Rearrange the equation to isolate \(\sqrt{y+2}\)
First, we want to isolate the term involving the square root. Let's move \(x - 3\) to one side and \(-1 + \sqrt{3}\) to the other side:
[tex]\[ \sqrt{y + 2} = -1 + \sqrt{3} - (x - 3) \][/tex]
[tex]\[ \sqrt{y + 2} = \sqrt{3} - x + 2 \][/tex]
### Step 2: Solve for \(x\)
To solve for \(x\), we can set it as:
[tex]\[ x = 3 - 1 + \sqrt{3} \][/tex]
[tex]\[ x = 2 + \sqrt{3} \][/tex]
### Step 3: Simplify the equation for \(y\)
Substitute \(x = 2 + \sqrt{3}\) back into the equation.
[tex]\[ \sqrt{y + 2} = \sqrt{3} - (2 + \sqrt{3}) + 2 \][/tex]
This simplifies to:
[tex]\[ \sqrt{y + 2} = \sqrt{3} - 2 - \sqrt{3} + 2 \][/tex]
[tex]\[ \sqrt{y + 2} = 0 \][/tex]
Square both sides to eliminate the square root:
[tex]\[ y + 2 = 0 \][/tex]
[tex]\[ y = -2 \][/tex]
### Step 4: Conclusion
The value of \(x\) is:
[tex]\[ x = 2 + \sqrt{3} \][/tex]
The value of \(y\) is:
[tex]\[ y = -2 \][/tex]
Thus, the final solution is:
[tex]\[
x = 3.73205080756888
\][/tex]
[tex]\[
y = -2
\][/tex]
