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 determine which equation represents a line that is perpendicular to the line x+2y=6, we need to find the slope of the line x+2y=6.
To put the equation in slope-intercept form (y=mx+b), we solve for y:
x + 2y = 6
2y = -x + 6
y = -0.5x + 3
The slope of this line is -0.5.
A line that is perpendicular to this line will have a slope that is the negative reciprocal of -0.5. The negative reciprocal of -0.5 is 2.
Using the point-slope form of a linear equation, y - y1 = m(x - x1), where (x1, y1) is the given point (3,-4) and m is the desired slope of 2:
y - (-4) = 2(x - 3)
y + 4 = 2x - 6
y = 2x - 6 - 4
y = 2x - 10
The equation that represents a line perpendicular to x+2y=6 and passes through the point (3, -4) is therefore:
C. −2x+y=−10