To find the point-slope form of a line with a given slope and a point that the line passes through, we use the point-slope formula of a linear equation:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Where:
- [tex]\( (x_1, y_1) \)[/tex] is a point on the line.
- [tex]\( m \)[/tex] is the slope of the line.
Given the slope [tex]\( m = 3 \)[/tex] and the point [tex]\((-1, 4)\)[/tex], we substitute these values into the point-slope formula.
First, identify the values for the slope [tex]\( m \)[/tex] and the coordinates [tex]\((x_1, y_1)\)[/tex]:
- Slope, [tex]\( m = 3 \)[/tex]
- Point, [tex]\( (x_1, y_1) = (-1, 4) \)[/tex]
Now, substitute these values into the formula:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substitute [tex]\( y_1 = 4 \)[/tex], [tex]\( m = 3 \)[/tex], and [tex]\( x_1 = -1 \)[/tex]:
[tex]\[ y - 4 = 3(x - (-1)) \][/tex]
This simplifies to:
[tex]\[ y - 4 = 3(x + 1) \][/tex]
Thus, the point-slope form of the line that has a slope of 3 and passes through the point [tex]\((-1, 4)\)[/tex] is:
[tex]\[ y - 4 = 3(x + 1) \][/tex]
Among the given options, this matches:
[tex]\[ y - 4 = 3[(x - (-1))] \][/tex]
So, the correct answer is:
[tex]\[ y - 4 = 3[(x - (-1))] \][/tex]
