Transcript Document

Writing Equations
of a Line
Subtitle: What is the
minimum information
needed?
Various Forms of an Equation of a
Line.
y  mx  b
Slope-Intercept Form
Standard Form
m  slope of the line
b  y  intercept
Ax  By  C
A, B, and C are integers
A  0, A must be postive
y  y1  m  x  x1 
Point-Slope Form
m  slope of the line
 x1 , y1  is any point
EXAMPLE 1 Write an equation given the slope and y-intercept
Write an equation of the line shown.
EXAMPLE 1 Write an equation given the slope and y-intercept
SOLUTION
3
From the graph, you can see that the slope is m =
4
and the y-intercept is b = –2. Use slope-intercept form
to write an equation of the line.
y = mx + b
Use slope-intercept form.
3
y = x + (–2)
4
3
y = x (–2)
4
3
Substitute for m and –2 for b.
4
Simplify.
GUIDED PRACTICE
for Example 1
Write an equation of the line that has the given slope
and y-intercept.
1.
m = 3, b = 1
ANSWER
y = 3x + 1
2.
m = –2 , b = –4
ANSWER
y = –2x – 4
m=–3 ,b=7
2
4
ANSWER
3.
y=– 3 x+ 7
2
4
EXAMPLE 2
Write an equation given the slope and a point
Write an equation of the line that passes
through (5, 4) and has a slope of –3.
SOLUTION
Because you know the slope and a point on the
line, use point-slope form to write an equation of
the line. Let (x1, y1) = (5, 4) and m = –3.
y – y1 = m(x – x1)
Use point-slope form.
y – 4 = –3(x – 5)
Substitute for m, x1, and y1.
y – 4 = –3x + 15
Distributive property
y = –3x + 19
Write in slope-intercept form.
EXAMPLE 3
Write equations of parallel or perpendicular lines
Write an equation of the line that passes through (–2,3)
and is (a) parallel to, and (b) perpendicular to, the line
y = –4x + 1.
SOLUTION
a.
The given line has a slope of m1 = –4. So, a line
parallel to it has a slope of m2 = m1 = –4. You know
the slope and a point on the line, so use the pointslope form with (x1, y1) = (–2, 3) to write an equation
of the line.
EXAMPLE 3
Write equations of parallel or perpendicular lines
y – y1 = m2(x – x1)
Use point-slope form.
y – 3 = –4(x – (–2))
Substitute for m2, x1, and y1.
y – 3 = –4(x + 2)
Simplify.
y – 3 = –4x – 8
Distributive property
y = –4x – 5
Write in slope-intercept form.
EXAMPLE 3
Write equations of parallel or perpendicular lines
b. A line perpendicular to a line with slope m1 = –4 has
a slope of m2 = – 1 = 1 . Use point-slope form with
m1 4
(x1, y1) = (–2, 3)
y – y1 = m2(x – x1)
1
y – 3 = (x – (–2))
4
1
y – 3 = (x +2)
4
1
1
y–3= x+
4
2
1
7
y  x
4
2
Use point-slope form.
Substitute for m2, x1, and y1.
Simplify.
Distributive property
Write in slope-intercept form.
GUIDED PRACTICE
4.
Write an equation of the line that passes through
(–1, 6) and has a slope of 4.
ANSWER
5.
for Examples 2 and 3
y = 4x + 10
Write an equation of the line that passes through
(4, –2) and is (a) parallel to, and (b) perpendicular
to, the line y = 3x – 1.
ANSWER
y = 3x – 14
EXAMPLE 4
Write an equation given two points
Write an equation of the line that passes through
(5, –2) and (2, 10).
SOLUTION
The line passes through (x1, y1) = (5,–2) and
(x2, y2) = (2, 10). Find its slope.
10 – (–2)
y2 – y1
=
m=
=
x2 – x1
2 –5
12
–3
= –4
EXAMPLE 4
Write an equation given two points
You know the slope and a point on the line, so use
point-slope form with either given point to write an
equation of the line. Choose (x1, y1) = (4, – 7).
y2 – y1 = m(x – x1)
Use point-slope form.
y – 10 = – 4(x – 2)
Substitute for m, x1, and y1.
y – 10 = – 4x + 8
Distributive property
y = – 4x + 8
Write in slope-intercept form.