LINES

1. SLOPE

2. GRAPHING LINEAR EQUATIONS

3. WRITINGLINEAR EQUATIONS

4. PARALLEL & PERPENDICULAR

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Slope of a Line

Slope (gradient) is a ratio of the change in y (vertical change) to the change in x ( horizontal change)

The slope, denoted by m, of the line through the points and Is defined as follows:

11, yx 22 , yx

●

● 11, yx

22 , yx

12 xx

12 yy

12

12

xx

yymslope

1,2

4,6

26

14

Y

X0

26

14

m

4

3m

)1,2(, 11 yx

)4,6(, 22 yx

62

41

m

4

3

4

3

m

)4,6(, 11 yx

)1,2(, 22 yx

Invers

0

3

-5 6

-3

a

b

c

d

Find the slope of each segment (a, b, c and d)!

5

32

1

6

32

1

6

35

3

d

c

b

a

Lines with positive slope rise to the right

Lines with negative slope fall to the right

k

h

4

6

4

8

Slope of line 3

2

6

4k

Find the slope of line k and h!

Slope of line 2

1

8

4h

Linear equations can be written in different forms : Standard form and slope-intercept form.

Form Equation Slope

Standard

Slope-interceptcbyax

cmxy b

a

m

Example : Equation Form Slope

032 yx

15 yx

xy 21

352 xy

142 xy

224 yx

standard

standard

standard

Slope-intercept

Slope-intercept

Slope-intercept

3

2

5

1

5

1

2

2

5

22

4

22

4

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Graphing Linear EquationsTo draw the line we need two point determine a line.We can find the X-intercept and Y-intercept.

Example : graph the line 2x + 3y = 12

To find X-intercept, let y = 0

12032 x

122 x

6x

Thus, (6 , 0) is a point on the line

To find Y-intercept, let x = 0

123)0(2 y

123 y

4y

Thus, (0 , 4) is a point on the line

Y

X0

● (0 , 4)

● (6 , 0)

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Writing Linear Equations

1. An equation of the line that passes through the point and has slope m is : 11, yx

11 xxmyy

Example : Find an equation of the line through (1 , 3) and its slope 2

3,1, 11 yxSolution :

2m

11 xxmyy

123 xy

223 xy

322 xy

12 xy

2. An equation of the line that passes through the point and is : 11, yx 22 , yx

12

1

12

1

xx

xx

yy

yy

Example : Find an equation of the line through (-1 , 4) and (2 , -3)

4,1, 11 yxSolution :

3,2, 22 yx)1(2

)1(

43

4

xy

3

1

7

4

xy

1743 xy

77123 xy

573 xy

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Parallel and Perpendicular Lines

2k2121 // mmkk

2h

1h

Parallel

Perpendicular

Two lines are parallel if and only if their slope equal

If slope k1 = m1 and slope k2 = m2

12121 mmhh

Two lines are Perpendicular if and only if the product ofTheir slope = -1

If slope h1 = m1 and slope h2 = m2

1k

Let slope second line is , then

Example : check the two lines parallel or perpendicular and 542 yx 036 yx

Solution : Let slope the first line is ,then 1m 2

1

4

21

m

2m 23

62 m

21 mm The line are not parallel

)2(2

121 mm

1 The line are Perpendicular

Exercise, check the two line parallel or perpendicular

123532.1 yxdanyx

1522.2 yxdanyx

13602.3 yxdanyx

221095.4 yxdanyx

02236.5 yxdanyx

parallelm

m

1

1

2

1

larperpendicum

m

2

2

2

1

larperpendicum

m

2

33

2

2

1

larperpendicum

m

55

1

2

1

larperpendicum

m

2

12

2

1

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