To find the equation of a line that passes through a given point and has a specific slope, we use the point-slope form of a line equation. 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.
Given:
- The point [tex]\((-2, -8)\)[/tex]
- The slope [tex]\(m = 3\)[/tex]
First, substitute the given point [tex]\((-2, -8)\)[/tex] and slope [tex]\(m = 3\)[/tex] into the point-slope form equation:
[tex]\[ y - (-8) = 3(x - (-2)) \][/tex]
Simplify the equation:
[tex]\[ y + 8 = 3(x + 2) \][/tex]
Next, distribute the slope [tex]\(m = 3\)[/tex]:
[tex]\[ y + 8 = 3x + 6 \][/tex]
Finally, isolate [tex]\(y\)[/tex] to convert this into the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[ y = 3x + 6 - 8 \][/tex]
[tex]\[ y = 3x - 2 \][/tex]
So, the equation of the line in slope-intercept form is:
[tex]\[ y = 3x - 2 \][/tex]
Among the given options, the correct equation of the line that passes through the point [tex]\((-2, -8)\)[/tex] and has a slope of 3 is:
[tex]\[ \boxed{y = 3x - 2} \][/tex]
