Equations of a Line

EQUATIONS OF A LINE

By: Mr. Xandro Alexi A. Nieto, M.Ed.Math UST Faculty of Pharmacy

run = 2

Review of Terms Before proceeding to our lesson, please recall the

following terms: Slope (m) rise / run from any point in the line. x-intercept (a) point of the x-axis where the line pass through.

To illustrate,

x-int (a) - 2

y-int (b) 1 rise = 1

y-intercept (b) point of the y-axis where the line pass through.

FORMS OF A LINE

The three forms of a line that we are going to discuss in this presentation are, as follows:

1by

a

x

slope-intercept form

point slope form two-point form

Some Analytic Geometry books would include the intercept form, . But since equations in those forms may be derived using the two-point form, it wont be discussed anymore in this presentation.

bm xy 11 xxmyy

112

121 xx

xx

yyyy

SLOPE-INTERCEPT FORM


The slope-intercept form is used when the given are slope (m) and y-intercept (b) of the line. Do not use the slope-intercept form if the given are slope (m) and x-intercept (a).

Any equation in standard form, is already in slope-intercept form.

bm xyslope y-intercept


EXAMPLES:

bm xy

Find the equation of the line with slope of and y-intercept of 3. 3

4

To graph the given, m = and b = - 3. 34

run = 3

rise = 4

y-int = -3

To find the equation of the line, use .

Since m = and b = -3, then 34

334 xy , then express in general form

0934943

394

yxxy

xy

is the equation of the line.


EXAMPLES:

bm xy

Find the equation of the line with slope of and y-intercept of .

32

To graph the given, m = and b = .

run = 3

rise = -2

y-int = 1/2

To find the equation of the line, use .

Since m = and b = , then

21

32 xy , then express in general form

0364346

634

yxxy

xy


32

32

21

21

21

TRY THIS! 41


Find the equation of the line with slope of and y-intercept of .

32

ANSWER: 3x + 12y - 8 = 0

POINT-SLOPE FORM

Let (x,y) be the unknown point and (x1,y1) be the given point.

By definition of slope ,

and since the (x,y) is an unknown point, then .

By cross-multiplication, .

POINT-SLOPE FORM The point-slope form is used when the given are slope

(m) and a point passing through the line (x1,y1). Suppose you want to find the slope of the line, in which theres only

one given point (x1,y1).

1

1

xx

yym

11 yyxxm

Thus, the formula were going to use is , where m is the given slope and (x1,y1) is the given point.

11 xxmyy

12

12

xx

yym

To find the equation, it is given that

m = - 3 and (x1,y1)=(1, 2).

Find the equation of the line with slope of - 3 and passing through (1, 2).

POINT-SLOPE FORM

11 xxmyy

EXAMPLES:

(1,2) rise = - 3

run = 1

Substitute the given to

132 xy332 xy

053 yx is the equation of the line.

To graph the given, note that the slope is -3 or -3/1. Thus, giving a rise of -3 and run of 1.

To find the equation, it is given that

m = and (x1,y1)=(-3,1).

Find the equation of the line with slope of and passing through (-3, 1).

POINT-SLOPE FORM

32

32

11 xxmyy

EXAMPLES:

(-3,1)

run = 3

rise = 2

Substitute the given to )3(

321 xy 3

321 xy

3213 xy6233 xy0932 yx is the equation of the line.

Find the equation of the line with slope of and passing through (-3, -2).

POINT-SLOPE FORM

41

TRY THIS!

ANSWER: x + 4y + 11 = 0

TWO-POINT FORM

TWO-POINT FORM

The two-point form is used when the given are two points ((x1,y1) & (x2,y2)) passing through the line.

11 xxmyy Weve learned in the point-slope form that and from the definition of slope that .

12

12

xx

yym

112

121 xx

xx

yyyy

Thus, by making m as , in the point-slope form, we have

12

12

xx

yy

where ((x1,y1) & (x2,y2)) are the two points passing through the line.

TWO-POINT FORM

EXAMPLES: Find the equation of the line passing through (1,2) and (4,3).

To graph, simply plot the two points then connect by drawing a line.

(4,3) (1,2)

To find the equation of the line, substitute (1,2) and (4,3) to (x1,y1) and (x2,y2)) respectively.

112

121 xx

xx

yyyy

114232

xy

1312 xy

1123 xy

163 xy053 yx is the equation of the line.

TWO-POINT FORM

EXAMPLES: Find the equation of the line passing through and .

(3,-0.5)

(-0.25,-2)

112

121 xx

xx

yyyy

41

413

221

2 xy

41

41323

2 xy

41

1362 xy

416213 xy

2362613 xy


2,41

21

,3

0249136 yx

0492612 yx

TWO-POINT FORM

TRY THIS! Find the equation of the line passing through and (1,-2).

41

,3

ANSWER: 9x + 16y + 23 = 0

REVIEW

So far, we have discussed the three forms of equations and we derived equations by direct substitution of the given.

slope-intercept form

point slope form two-point form

bm xy 11 xxmyy

112

121 xx

xx

yyyy

WHAT IFs?

What if the given are the slope and the x-intercept of the line?

What if the given are a point passing through the line and x or y-intercept?

What if the given are the lines x and y-intercepts?

Lets answer these questions one by one.


Do not use the slope-intercept form because the slope-intercept form is used if the given are the slope and the y-intercept (NOT THE X-INTERCEPT!).


Instead, recall that any x-intercept (a) may be expressed as ordered pair (a,0). Thus, you may use point-slope form.

What if the given are the slope and the x-intercept of the line? EXAMPLE:

Find the equation of the line with slope of 3 and has an x-intercept of 2.

run = 1

rise = 3

Since the x-intercept is 2, then its ordered pair is (2,0). Considering the slope of 3, then we have to use the point-slope form.

06363

23011

yxxy

xyxxmyy


x-intercept = 2

What if the given are a point passing through the line and x or y-intercept? Recall that any x-intercept (a) may be expressed in

ordered pair (a,0).

Likewise, any y-intercept (b) may be expressed in ordered pair (0,b).

Thus, the two-point form may be used in cases like this.

What if the given are a point passing through the line and x or y-intercept? EXAMPLE:

Find the equation of the line passing through (3,-1) and having a y-intercept of -2.

(3,-1) Y-intercept = - 2

Since the y-intercept is -2, then its ordered pair is (0,-2). Considering the other given, (3,-1), then we can make use of the two-point form.

063

6323

312

003

212

112

121

yxxy

xy

xy

xy

xxxx

yyyy



Again, recall that any x-intercept (a) may be expressed in ordered pair (a,0).

Likewise, any y-intercept (b) may be expressed in ordered pair (0,b).

Thus, the two-point form may be used in cases like this.


EXAMPLE: Find the equation of the line having an x-intercept of 3 and a y-intercept of 2?

y-intercept = 2

x-intercept = 3

Since the x-intercept is 3, then its ordered pair is (3,0). Likewise, since the y-intercept is 2, then its ordered pair is (0,2). Thus, we can now make use of the two-point form.

0632623

33

2

330020

112

121

yxxy

xy

xy

xxxx

yyyy


Were done with the topic.

You may now answer the 10-point online assignment,

linear equations in two variables.

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Equations of a Line