To find the equation of the line passing through the point [tex]\((-5, 1)\)[/tex] with a slope of [tex]\(\frac{3}{2}\)[/tex], we can use the point-slope form of the equation of a line. The point-slope form is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line, and [tex]\(m\)[/tex] is the slope of the line. In this problem, the point [tex]\((-5, 1)\)[/tex] is given along with the slope [tex]\(m = \frac{3}{2}\)[/tex].
Plugging in the given point and slope into the point-slope form equation, we have:
[tex]\[ y - 1 = \frac{3}{2}(x - (-5)) \][/tex]
Since subtracting a negative number is equivalent to adding the positive counterpart, we can simplify the equation:
[tex]\[ y - 1 = \frac{3}{2}(x + 5) \][/tex]
Therefore, the correct equation of the line is:
[tex]\[ y - 1 = \frac{3}{2}(x + 5) \][/tex]
Comparing this with the options provided:
1. [tex]\(y - 5 = \frac{3}{2}(x + 1)\)[/tex]
2. [tex]\(y + 1 = \frac{3}{2}(x - 5)\)[/tex]
3. [tex]\(y + 5 = \frac{3}{2}(x - 1)\)[/tex]
4. [tex]\(y - 1 = \frac{3}{2}(x + 5)\)[/tex]
We see that option 4 matches our derived equation. Hence, the correct choice is:
[tex]\[ \boxed{y - 1 = \frac{3}{2}(x + 5)} \][/tex]
