Question

find the real solution of the following radical equation.

\\\sqrt{y - 19} = 2\\

\\y = ?\\

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Solving Radical Equations"
],
"new_concepts": [],
"current_concepts": [
"Radical Equations",
"Solving Radical Equations"
]
}
</pre_analysis>

<reasoning>

Square both sides of the equation

\[

$$\begin{aligned} (\sqrt{y - 19})^2 &= 2^2 \\ y - 19 &= 4 \end{aligned}$$

\]

Solve for y

\[

$$\begin{aligned} y &= 4 + 19 \\ y &= 23 \end{aligned}$$

\]

Verify the solution

\[

$$\begin{aligned} \sqrt{23 - 19} &= \sqrt{4} = 2 \end{aligned}$$

\]
</reasoning>

<answer>
Find the real solution of the following radical equation.
\[\sqrt{y - 19} = 2\]
\(y =\) <blank>23</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Solving Radical Equations"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Solving Radical Equations"
],
"new_concepts": [],
"current_concepts": [
"Radical Equations",
"Solving Radical Equations"
]
}
</pre_analysis>

<reasoning>

Square both sides of the equation

\[

$$\begin{aligned} (\sqrt{y - 19})^2 &= 2^2 \\ y - 19 &= 4 \end{aligned}$$

\]

Solve for y

\[

$$\begin{aligned} y &= 4 + 19 \\ y &= 23 \end{aligned}$$

\]

Verify the solution

\[

$$\begin{aligned} \sqrt{23 - 19} &= \sqrt{4} = 2 \end{aligned}$$

\]
</reasoning>

<answer>
Find the real solution of the following radical equation.
\[\sqrt{y - 19} = 2\]
\(y =\) <blank>23</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Solving Radical Equations"
]
}
</post_analysis>