Which of the following equations represents a line that is perpendicular to the line represented by x + 2y = 6 and passes through the point (3, −4)?
A. 2x + y = 2
B. x + 2y = 11
C. −2x + y = −10
D. −2x + y = 7
To find the equation of a line that is perpendicular to the line x + 2y = 6, we need to determine the slope of the given line and then find the negative reciprocal of that slope.
First, let's rewrite the given equation in slope-intercept form, y = mx + b, where m is the slope:
x + 2y = 6
2y = -x + 6
y = -1/2x + 3
The slope of the given line is -1/2.
The negative reciprocal of -1/2 is 2.
Now, let's substitute the point (3, -4) into the equation y = mx + b to find the y-intercept (b):
-4 = 2(3) + b
-4 = 6 + b
b = -10
Therefore, the equation of the line that is perpendicular to x + 2y = 6 and passes through the point (3, -4) is: y = 2x - 10
Which is the same as option C. Therefore, the correct answer is C. −2x + y = −10