Answer:
[tex]\displaystyle y = -4x - 5[/tex]
Step-by-step explanation:
We want to find the slope-intercept form of the equation that passes through the point (-2, 3) and is perpendicular to the line:
[tex]\displaystyle y = \frac{1}{4} x - 4[/tex]
Note that this line has a slope of 1/4.
Recall that the slopes of perpendicular lines are negative reciprocals of each other.
Since the slope of our old line is 1/4, the slope of its perpendicular line must be -4.
We are also given that it passes through the point (-2, 3). So, we can consider using point-slope form:
[tex]\displaystyle y - y_1 = m(x - x_1)[/tex]
Let (-2, 3) be (x₁, y₁) and substitute -4 for the slope m. Hence:
[tex]\displaystyle y - (3)= -4 (x - (-2))[/tex]
Convert into slope-intercept form. Simplify:
[tex]\displaystyle \begin{aligned} y -3 &= -4 (x + 2) \\ y - 3 &= -4x - 8 \\ y &= -4x -5\end{aligned}[/tex]
In conclusion, the perpendicular line that passes through the point (-2, 3) is given by:
[tex]\displaystyle y = -4x - 5[/tex]
