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8/13/2019 Writing Equations for Lines
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THE LEARNING CENTER Trident Technical College
WRITING EQUATIONS FOR LINES
12
12
xx
yy
= This is the formula for finding the !lo"eof a line that passes
through two points.
When can I #!e thi! $or#la% When I am given two points through
which a line passes and I am asked to find the slope of that
line
Exa"le &' Find the slope of the line which passes through
the points (2, -! and (-", -1!.
(
)
(
)
(
(&
*+
,(-&
xx
yy
&*
&* =
=
=
= An!.er# The slope of the line is
(
) .
This formula is called the !lo"e/interce"t $or#lafor a line
$ecause it displa%s $oththe slope and the %-intercept of the
line.
% & ' 0)lope & %-intercept & 0
When can I #!e thi! $or#la% When I am given the e*uation for a
line and I am asked to find the slope and %-intercept of that
line
or When I am given the slope and %-intercept of a line and I am
asked to write an e*uation for the line
Exa"le *' Find the slope and the %-intercept of the line whose
e*uation is 2' - +% & ".
To find the slope and %-intercept, solve the e*uation for %.
)olving for % automaticall% puts the e*uation into % & m' $
form, theform where %ou can readil% see the slope and the
%-intercept.
An!.er# The slope of the line is1
&. The %-intercept is -23 /+45,. 5
+x
1
&y
5
+
5
x*
5
y5
+x*y5
+y5x*
=
=
)lope %-intercept
Exa"le 6' Write an e*uation for the line that has a slope of 2
and a %-intercept of 12.
/emem$er that the numerical value of the slope replaces and the
numerical value of the %-interceptreplaces 0in % & ' 0.
An!.er# &*x*
7y .
/ev. 110"
Cl#e! a0o#t .hen to #!e di$$erent $or#la!
0xy
Cl#e! $or8AT &2*Serie!
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(2,!
(2,0!(2,-2!
(-,2! (0,2! (2,2!
WRITING EQUATIONS FOR LINES 9age *
This formula is called the "oint/!lo"e $or#lafor a line $ecause
m representsthe slope and '1and %1represent the '- and
%-coordinates of a point throughwhich the line passes.
When can I #!e thi! $or#la% When I am given the slope of a line
and a point and I am asked to write an e*uation for the line which
hasthe given slope and which passes through the given point
Exa"le 1' Write an e*uation for the line that passes through the
point (-", ! and that has a slope of ".
The slope of the line is m. m & "
The '-coordinate of the given point is '1. '1 %1The %-coordinate
of the given point is %1. (-", !
To get the e*uation for the line, plug in the numerical values
for m, '1, and %1Then isolate % to get the answer in % & m' $
form.
An!.er#&2x
+
(y
6(x
+
(y
(x+
(6y
,+x-+
(6y
,,+-x-+
(6y
,xx-yy&&
:ertical Line!The slope of all vertical lines is undefined.
;
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WRITING EQUATIONS FOR LINES 9age 6
9arallel line!
If two lines are "arallel, their slopes arethe !ae numerical
value?
If the slope of a given line is ", then ever% line that is
parallel tothe given line will also have a slope of " $ecause
parallel lines
have the same slope.
9er"endic#lar line!
If two lines are "er"endic#lar, thenumerical values of their
slopes are thenegati>e reci"rocal!of each other.
If the slope of a given line is ", then ever% line that
isperpendicular to the given line will have a slope of ", $ecause"
and " are the negative reciprocals of each other.5egative means
that the slopes have opposite signs. /eciprocalmeans that the
numerator and denominator have traded places.(The numerator and
denominator have flipped upside down.)
Exa"le +' Write an e*uation for the line that is parallel to "'
2% & and that passes through the point(, 7!.
What strateg% should $e used8
In order to write an e*uation for a line, %ou usuall% need to
know the slope of that line. 3ou can use theslope of "' 2% &
since it is parallel to the line whose e*uation %ou are asked to
write. To find the slope of "' 2% & . solve for %. This puts
the e*uation into % & m' $ form. 9lug $oth the slope that %ou
:ust found and the given point into the point-slope formula.
point (, 7!'1 %1
)lope
An!.er'
Cl#e! a0o#t "arallel and "er"endic#lar line!
9#tting all the cl#e! together
*
7x
*
+y
7x+y*
7y*x+
=
*
*x
*y
&+x
*
+y
*
&x
*
+)y
6x-*
+)
xxyy &&
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An!.er'
point (, 2! '1 %1
9oints (", -1! (-;, -2!
'1 %1 '2 %2
WRITING EQUATIONS FOR LINES 9age 1
Exa"le )' Write an e*uation for the line that is perpendicular
to ' - % & 11 and that passes through thepoint (, 2!.
What strateg% should $e used8
In order to write an e*uation for a line, %ou usuall% need to
know the slope of that line. 3ou can use thenegative reciprocal of
the slope of ' - % & 11 since it is perpendicular to the line
whose e*uation %ou are
asked to write. To find the slope of ' - % & 11, solve for
%. This puts the e*uation into % & m' $ form. 9lug $oth the
negative reciprocal of the slope that %ou :ust found and the given
point into the point-slopeformula.
)lope
The negative reciprocal of 1 is .
Exa"le (' Write an e*uation for the line that passes through the
points (", -1! and (-;, -2!.
What strateg% should $e used8
In order to write an e*uation for a line, %ou usuall% need to
know the slope of that line. 3ou can use theslope formula to find
the needed slope. 5e't %ou can plug the slope and one of the given
points into the point-slope formula to get the e*uation forthe
line.
)lope
Exa"le 5' Write an e*uation for the line that passes through the
points (", ;! and (-, ;!.
What strateg% should $e used8
5otice that $oth points have the same %-value. /emem$er that all
of the points on a horiontal line havethe same %-values. Therefore,
these points are on a horiontal line. /emem$er that the e*uation
for a horiontal line can $e written as % & a constant. In this
case, theconstant must $e ; since that is the %-value common to
$oth points.
WRITING EQUATIONS FOR LINES 9age +
6
&&x
6
&y
6
&&
6
x
6
y6
&&xy6
yx
=
=
*x6
**&x6y
*&x6*y
(x-6*
,xx-yy&&
7
7
&
7
&*
+1
&-*
x
yy
&*
&*
=
=
=
=
=
x7
&y
7x
7
&
x7
&&
x-&y
+x-7
&
,&-
xy&
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Exa"le 7' Write an e*uation for the line that passes through the
points (", ;! and (", -7!.
What strateg% should $e used8
5otice that $oth points have the same '-value. /emem$er that all
of the points on a vertical line have thesame '-values. Therefore,
these points are on a vertical line. /emem$er that the e*uation for
a vertical line can $e written as ' & a constant. In this case,
the constantmust $e " since that is the '-value common to $oth
points.
An!.er' x ; +
9ractice "ro0le!'
1. Find an e*uation for the line that passes through the points
(;, 12! and (;, !. (Hint: Both points have thesame x-values.)
2. Write the e*uation for aline with a slope of 7" that passes
through the point (-, -1!.. Write an e*uation for the line which
passes through the point (,12! and which is parallel to ;' 2% &
7.(Hint: Parallel lines have the same slope.)
;. Find an e*uation for the line which passes through the points
(, -! and (-, !.". Write an e*uation for a line which passes
through the points (1, ;! and (1, 1!. (Hint: Both points have
the
same x-values.)7. Find an e*uation for the line which is
perpendicular to ' % & " and which passes through the point
(-7, !. (Hint: Perpendicular slopes are the neative reciprocals
of each other.). Write an e*uation for the line which passes
through the points (-, -"! and (;, -"!. (Hint: Both points have
the same !-values.)+. Find an e*uation for the line which has a
%-intercept of ; and which has a slope of .. Write an e*uation for
the line which passes through the point (-11, -! and which is
parallel to % & +. (Hint:
" # $ is a hori%ontal line. &ince all hori%ontal lines are
parallel to each other' the euation that !ou writeshould e a
hori%ontal line.)10. Write an e*uation for the line which passes
through the point (-11, -! and which is parallel to ' & 7.
(Hint:
x # * is a vertical line. &ince all vertical lines are
parallel to each other' the euation that !ou write shoulde a
vertical line.)
11. Write an e*uation for the line which passes through the
point (, +! and which is perpendicular to % & -.(Hint: +ertical
lines and hori%ontal lines are perpendicular to each other.
&ince ! # -, is a hori%ontal line'!ou will want to write an
euation for a vertical line.)
12. Write an e*uation for the line which passes through the
point (;, "! and which is perpendicular to ' & .(Hint: +ertical
lines and hori%ontal lines are perpendicular to each other.
&ince x # is a vertical line' !ouwill want to write an euation
for a hori%ontal line.)
An!.er! to "ractice "ro0le!'
;=
=
=
