2.4 Write Equations of Lines

 

Point-slope form

The point-slope form is given by y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

 

WRITING AN EQUATION OF A LINE

Use slope-intercept form: Given slope m and y-intercept b, use the equation

y = __mx+ b_.

Use point-slope form: Given slope m and a point (x1, y1) use the equation

y - y1 =_m(x - x1)_.

Use two points: Given points (x1, y1) and (x2, y2), first use the __slope__ formula to find m. Then use the __point-slope__ form with either given point.

 

Example 1

Write an equation given the slope and y-intercept

 

Write an equation of the line shown.

Solution

From the graph, you can see that the slope is m =

_____

and the y-intercept is b = __1__. Use the slope intercept form to write an equation of the line.

 

 


y = mx + b

Use slope-intercept form.

y =         + _1_

____  

Substitute          for m and __1__ for b.

____

 

 

 

Example 2

Write an equation given the slope and a point

 

Write an equation of the line that passes through (2,1) and has a slope of 2.

 

Solution

Because you know the slope and a point on the line, use the point-slope form to write an equation of the line. Let (x1, y1) = ( _2_, _1_ ) and m = _2_.

y - y1 = m(x – x1)

Use point-slope form.

y - _1_ = _2_ (x - _2_ )

Substitute for m, x1, and y1.

y - _1_ = _2_ x - _4_

Distributive property

y = _2_ x - _3_

Write in slope-intercept form.

 

Example 3

Write equations of parallel or perpendicular lines

 

Write an equation of the line that passes through (-1, 1) and is (a) parallel to, and (b) perpendicular to, the line y = -2x + 3.

 

Solution

The given line has a slope of m1 = _-2_. A line that is parallel to it must have a slope of m2 = m1 = _-2_. A line perpendicular to a line with slope m1 = _-2_ must have a slope of m2 =                            Use the point-slope form with (x1, y1) = (_-1_, _1_) to write an equation of the line.

a.           y – y1 = m2(x - x1)

y - _1_ = _-2_ (x - _(-1)_)

y = _-2_ x - _1_

b.        y - y1 = m 2(x – x1)

y - _1_ =          (x - _(-1)_)

___

y =      x +

___     ___

 

 

Example 4

Write an equation given two points

 

Write an equation of the line through (3, 1) and (2, -3).

The line passes through (x1, y1) = (3, 1) and (x2, y2) = (2, -3). Find its slope.

m =_4_

Use the point-slope form with either given point to write an equation of the line.

y – y1 = m(x – x1)

Use point-slope form.

_y - 1_ = _4(x- 3)_

Substitute for m, x1, and y1.

y = _4x - 11_

Write in slope-intercept form.