1) The document discusses various forms of equations for lines, including slope-intercept form, standard form, and point-slope form. It provides definitions and examples of writing equations of lines given the slope and y-intercept or given two points on the line.
2) Key concepts covered include writing the equation of a line given its slope m and y-intercept b using slope-intercept form y=mx+b, or given slope m and a point (x1,y1) using point-slope form y-y1=m(x-x1).
3) Examples are provided for writing equations of lines using slope-intercept form when given slope and y-intercept, and using point-
2. Various Forms of an Equation of a
Line.
Slope-Intercept Form
Standard Form
Point-Slope Form
slope of the line
intercept
y mx b
m
b y
= +
=
= β
, , and are integers
0, must be postive
Ax By C
A B C
A A
+ =
>
( )
( )
1 1
1 1
slope of the line
, is any point
y y m x x
m
x y
β = +
=
-
-
3. KEY CONCEPT
Writing an Equation of a Line
β Given slope m and y-intercept b
β’ Use slope-intercept form y=mx+b
β Given slope m and a point (x1,y1)
β’ Use point-slope form
β y - y1 = m ( x β x1)
β’ Given points (x1,y1) and (x2,y2)
β Find your slope then use point-slope form with either point.
4. Write an equation given the slope and y-interceptEXAMPLE 1
Write an equation of the line shown.
5. SOLUTION
Write an equation given the slope and y-interceptEXAMPLE 1
From the graph, you can see that the slope is m =
and the y-intercept is b = β2. Use slope-intercept form
to write an equation of the line.
3
4
y = mx + b Use slope-intercept form.
y = x + (β2)
3
4
Substitute for m and β2 for b.
3
4
y = x β2
3
4
Simplify.
6. GUIDED PRACTICE for Example 1
Write an equation of the line that has the given slope
and y-intercept.
1. m = 3, b = 1
y = x + 13
ANSWER
2. m = β2 , b = β4
y = β2x β 4
ANSWER
3. m = β , b =3
4
7
2
y = β x +3
4
7
2
ANSWER
7. Write an equation given the slope and a pointEXAMPLE 2
Write an equation of the line that passes
through (5, 4) and has a slope of β3.
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 = β3.
y β y1 = m(x β x1) Use point-slope form.
y β 4 = β3(x β 5) Substitute for m, x1, and y1.
y β 4 = β3x + 15 Distributive property
SOLUTION
y = β3x + 19 Write in slope-intercept form.
8. EXAMPLE 3
Write an equation of the line that passes through (β2,3)
and is (a) parallel to, and (b) perpendicular to, the line
y = β4x + 1.
SOLUTION
a. The given line has a slope of m1 = β4. So, a line
parallel to it has a slope of m2 = m1 = β4. You know
the slope and a point on the line, so use the point-
slope form with (x1, y1) = (β2, 3) to write an equation
of the line.
Write equations of parallel or perpendicular lines
9. EXAMPLE 3
y β 3 = β4(x β (β2))
y β y1 = m2(x β x1) Use point-slope form.
Substitute for m2, x1, and y1.
y β 3 = β4(x + 2) Simplify.
y β 3 = β4x β 8 Distributive property
y = β4x β 5 Write in slope-intercept form.
Write equations of parallel or perpendicular lines
10. EXAMPLE 3
b. A line perpendicular to a line with slope m1 = β4 has
a slope of m2 = β = . Use point-slope form with
(x1, y1) = (β2, 3)
1
4
1
m1
y β y1 = m2(x β x1) Use point-slope form.
y β 3 = (x β (β2))
1
4
Substitute for m2, x1, and y1.
y β 3 = (x +2)
1
4
Simplify.
y β 3 = x +
1
4
1
2
Distributive property
Write in slope-intercept form.
Write equations of parallel or perpendicular lines
1 7
4 2
y x= +
11. GUIDED PRACTICE for Examples 2 and 3GUIDED PRACTICE
4. Write an equation of the line that passes through
(β1, 6) and has a slope of 4.
y = 4x + 10
5. Write an equation of the line that passes through
(4, β2) and is (a) parallel to, and (b) perpendicular
to, the line y = 3x β 1.
y = 3x β 14ANSWER
ANSWER
12. Write an equation given two pointsEXAMPLE 4
Write an equation of the line that passes
through (5, β2) and (2, 10).
SOLUTION
The line passes through (x1, y1) = (5,β2) and
(x2, y2) = (2, 10). Find its slope.
y2 β y1
m =
x2 β x1
10 β (β2)
=
2 β 5
12
β3
= = β4
13. Write an equation given two pointsEXAMPLE 4
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) = (2, 10).
y2 β y1 = m(x β x1) Use point-slope form.
y β 10 = β 4(x β 2) Substitute for m, x1, and y1.
y β 10 = β 4x + 8 Distributive property
Write in slope-intercept form.y = β 4x + 8