To find the equation of the line that passes through the point [tex]\((-1, -1)\)[/tex] and has a slope of [tex]\(-4\)[/tex], we can utilize 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.
Here, we have:
[tex]\[ (x_1, y_1) = (-1, -1) \][/tex]
[tex]\[ m = -4 \][/tex]
Substitute the given point and slope into the point-slope form:
[tex]\[ y - (-1) = -4(x - (-1)) \][/tex]
[tex]\[ y + 1 = -4(x + 1) \][/tex]
Next, we need to simplify this equation into slope-intercept form [tex]\( y = mx + b \)[/tex]:
Expand the right-hand side:
[tex]\[ y + 1 = -4(x + 1) \][/tex]
[tex]\[ y + 1 = -4x - 4 \][/tex]
Isolate [tex]\( y \)[/tex] by subtracting 1 from both sides:
[tex]\[ y = -4x - 4 - 1 \][/tex]
[tex]\[ y = -4x - 5 \][/tex]
Therefore, the equation of the line in slope-intercept form is:
[tex]\[ y = -4x - 5 \][/tex]
