To find the equation of a line in slope-intercept form that passes through a given point and has a specified slope, we use the slope-intercept form equation, which is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
We have the following information:
- The point [tex]\((-8, -1)\)[/tex], which means [tex]\( x_1 = -8 \)[/tex] and [tex]\( y_1 = -1 \)[/tex].
- The slope [tex]\( m = -\frac{5}{4} \)[/tex].
First, we substitute the coordinates of the point and the slope into the general slope-intercept form equation to solve for [tex]\( b \)[/tex]:
[tex]\[ y_1 = mx_1 + b \][/tex]
Substituting [tex]\( x_1 = -8 \)[/tex], [tex]\( y_1 = -1 \)[/tex], and [tex]\( m = -\frac{5}{4} \)[/tex]:
[tex]\[ -1 = \left( -\frac{5}{4} \right)(-8) + b \][/tex]
Next, we simplify and solve for [tex]\( b \)[/tex]:
[tex]\[ -1 = \left( -\frac{5}{4} \right)(-8) + b \][/tex]
[tex]\[ -1 = \frac{40}{4} + b \][/tex]
[tex]\[ -1 = 10 + b \][/tex]
To isolate [tex]\( b \)[/tex], we subtract 10 from both sides of the equation:
[tex]\[ -1 - 10 = b \][/tex]
[tex]\[ b = -11 \][/tex]
Now that we have the value of the y-intercept [tex]\( b = -11 \)[/tex], we substitute [tex]\( m \)[/tex] and [tex]\( b \)[/tex] back into the slope-intercept form equation to get the final equation of the line:
[tex]\[ y = -\frac{5}{4}x - 11 \][/tex]
Thus, the equation of the line in slope-intercept form is:
[tex]\[ y = -\frac{5}{4}x - 11 \][/tex]
