To write the equation of a line perpendicular to the given equation, we first need to determine its slope. After that, we will write a general equation and use the given point to determine the y-intercept.
Calculating the Perpendicular Line's Slope
Two lines are perpendicular when their slopes are negative reciprocals. This means that the product of a given slope and the slope of a line perpendicular to it will be -1.
m_1*m_2=-1
For any equation written in slope-intercept form, y=mx+ b, we can identify its slope as the value of m.
Since the given equation is not written in slope-intercept form, we have to rewrite it before identifying the slope.
Looking at the given equation, we can see that its slope is 12.
y=1/2x -2
By substituting this value into our negative reciprocal equation for m_1, we can solve for the slope of a perpendicular line, m_2.
Any line perpendicular to the given equation will have a slope of - 2.
Writing the Perpendicular Line's Equation
Using the slope m_2=- 2, we can write a general equation in slope-intercept form for all lines perpendicular to the given equation.
y=- 2x+b
By substituting the given point ( 1, - 6) into this equation for x and y, we can solve for the y-intercept b of the perpendicular line.
Now that we have the y-intercept, we can complete the equation. The line given by this equation is both perpendicular to y= 12x-2 and passes through the point (1,- 6).
y=- 2x+(- 4) ⇔ y = - 2x - 4
