Write an equation in standard form of the line that contains the point (-2,4) and is
a. parallel to the line y=5x+12
b. perpendicular to the line y=5x+12
a. Which of the following equations, written in standard form, is parallel to the line y=5x+12 and contains the point (-2,4)? Choose the correct answer below.
a. 5x-y=-14
b. x+y=6/5
c. x-y=-6/5
d. 5x+y=14
To find the equation of a line parallel to another line, we need to use the same slope. The equation y = 5x + 12 is already in slope-intercept form (y = mx + b), where the slope is 5 and the y-intercept is 12. Therefore, the desired line will also have a slope of 5.
Using the point-slope form of a linear equation, which is y - y1 = m(x - x1), we can substitute the given point (-2, 4) and the slope 5 into the equation to find the equation for the parallel line:
y - 4 = 5(x - (-2))
y - 4 = 5(x + 2)
y - 4 = 5x + 10
y = 5x + 14
To write this equation in standard form, we rearrange the terms:
5x - y = -14
So, the correct answer for part a is (d) 5x - y = -14.
b. To find the equation of a line perpendicular to another line, we first need to determine the negative reciprocal of the slope. The slope of the given line y = 5x + 12 is 5, so the negative reciprocal is -1/5.
Using the same point-slope form as before and substituting the given point (-2, 4) and the slope -1/5:
y - 4 = (-1/5)(x - (-2))
y - 4 = (-1/5)(x + 2)
y - 4 = (-1/5)x - 2/5
Rearranging terms:
(1/5)x + y = 4 + 2/5
(1/5)x + y = 22/5
To write this equation in standard form, we multiply all terms by 5 to eliminate the fraction:
x + 5y = 22
So, the correct answer for part b is x + 5y = 22.