To convert the given point-slope form equation [tex]\(y - 2 = 3(x + 1)\)[/tex] into the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept, follow these steps:
1. Start with the point-slope form equation:
[tex]\[ y - 2 = 3(x + 1) \][/tex]
2. Distribute the slope (3) to the terms inside the parenthesis:
[tex]\[ y - 2 = 3x + 3 \][/tex]
3. Isolate [tex]\(y\)[/tex] by adding [tex]\(2\)[/tex] to both sides of the equation:
[tex]\[ y = 3x + 3 + 2 \][/tex]
4. Simplify the right side of the equation:
[tex]\[ y = 3x + 5 \][/tex]
So, the slope-intercept form of the equation [tex]\(y - 2 = 3(x + 1)\)[/tex] is [tex]\(y = 3x + 5\)[/tex].
Therefore, the correct choice from the given options:
[tex]\[
\begin{array}{l}
1.\quad y = 3x + 1 \\
2.\quad y = 3x - 3 \\
3.\quad y = 3x + 5
\end{array}
\][/tex]
is:
[tex]\[ y = 3x + 5 \][/tex]
So, the correct option is:
[tex]\[ \boxed{3} \][/tex]
