To determine the equation of a line through the point [tex]\((6, -2)\)[/tex] with slope [tex]\(m = 3\)[/tex], we use the slope-intercept form of a line, which is given as:
[tex]\[ y = mx + b \][/tex]
Here:
- [tex]\(m\)[/tex] is the slope of the line.
- [tex]\(b\)[/tex] is the y-intercept, which is the y-coordinate of the point where the line crosses the y-axis.
We are given:
- The coordinates of the point [tex]\( (x_1, y_1) = (6, -2) \)[/tex]
- The slope [tex]\( m = 3 \)[/tex]
We need to find the value of [tex]\(b\)[/tex] (the y-intercept). To do this, we substitute the coordinates of the given point and the slope into the slope-intercept form equation and solve for [tex]\(b\)[/tex].
Given:
[tex]\[ y_1 = mx_1 + b \][/tex]
Substitute [tex]\(x_1 = 6\)[/tex], [tex]\(y_1 = -2\)[/tex], and [tex]\(m = 3\)[/tex]:
[tex]\[ -2 = 3(6) + b \][/tex]
Simplify and solve for [tex]\(b\)[/tex]:
[tex]\[ -2 = 18 + b \][/tex]
Subtract 18 from both sides:
[tex]\[ -2 - 18 = b \][/tex]
[tex]\[ b = -20 \][/tex]
Now, we have the value of [tex]\(b\)[/tex]. Therefore, the equation of the line in slope-intercept form [tex]\( y = mx + b \)[/tex] is:
[tex]\[ y = 3x - 20 \][/tex]
So, the final answer is:
[tex]\[ \boxed{y = 3x - 20} \][/tex]
