Answer:
[tex]\displaystyle y = \frac{1}{2} x - 7[/tex]
Step-by-step explanation:
We want to determine the equation of the line that passes through the point (6, -4) and is perpendicular to the line:
[tex]\displaystyle y = -2x - 3[/tex]
Recall that the slopes of perpendicular lines are negative reciprocals of each other.
The slope of the given line is -2.
Hence, the slope of the perpendicular line is 1/2.
Therefore, the slope of our new line is 1/2. We also know that it passes through the point (6, -4). Since we are given the slope and a point, we can consider using point-slope form:
[tex]\displaystyle y - y_1 = m(x - x_1)[/tex]
Where m is the slope and (x₁, y₁) is a point.
Let (6, -4) be (x₁, y₁). Substitute:
[tex]\displaystyle y - (-4) = \frac{1}{2}(x - (6))[/tex]
Simplify. Distribute:
[tex]\displaystyle y + 4 = \frac{1}{2} x - 3[/tex]
And subtract 4 from both sides:
[tex]\displaystyle y = \frac{1}{2} x - 7[/tex]
In conclusion, the equation of our line is:
[tex]\displaystyle y = \frac{1}{2} x - 7[/tex]
