To write the equation of a line perpendicular to the one whose equation is given, we first need to determine its slope.
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. Looking at the given equation, we can see that its slope is 2.
y=2x+1
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 - 12.
Writing the Perpendicular Line's Equation
Using the slope m_2=- 12, we can write a general equation in slope-intercept form for all lines perpendicular to the given equation.
y=-1/2x+b
By substituting the given point P( 2, 8) 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=2x+1 and passes through the point P(2,8).
y=-1/2x+9
Finally, we can verify our answer by graphing both lines on the same coordinate plane. If they are perpendicular, they will intersect at a right angle.
We can see by looking at the graphs of the functions that they are indeed perpendicular.
