a To write the equation of a line perpendicular to the given equation, 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 - 25.
y=-2/5x+ 6
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 52.
Writing the Perpendicular Line's Equation
Using the slope m_2=52, we can write a general equation in slope-intercept form for all lines perpendicular to the given equation.
y=5/2x+b
By substituting the given point ( 2, - 3) 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=- 25x+6 and passes through the point (2,- 3).
y=5/2x+(- 8) ⇔ y=5/2x-8
b When lines are parallel, they have the same slope.To help us identify the slope of this line, let's first convert it into slope-intercept form, y=mx+ b, where m is the slope and (0, b) is the y-intercept.
With this, we can more easily identify the slope m and y-intercept b.
y=3/2x+ 5
This means we can write a general equation in slope-intercept form for all lines parallel to the given equation.
y=3/2x+ b
We are asked to write the equation of a line parallel to the given equation that passes through the given point ( 4, 7). By substituting this point into the general equation for x and y, we will be able to solve for the y-intercept b of the parallel line.
Now that we have the y-intercept, we can conclude that the line given by the following equation is parallel to - 3x+2y=10 and passes through (4,7).
y=3/2x+ 1
