Linear Equations Equations of Lines. 7/2/2013 Linear Equations 2 Lines and Equations Point-Slope Form Given line L and point (x 1, y 1 ) on L Let (x,

Linear Equations

Equations of Lines

Linear Equations 27/2/2013

Lines and Equations Point-Slope Form

Given line L and point (x1, y1) on L

Let (x, y) be any other point on L

Find slope

y

x

(x1, y1)

(x, y)L

∆x = x – x1

∆y = y – y1 m =

∆y∆x


Lines and Equations Point-Slope Form

y

x

L

∆x = x – x1

∆y = y – y1

m =∆y∆x

=y – y1 x – x1

Point-Slope Form

y – y1 = m(x –x1) (x1, y1)

(x, y)


Slope-Intercept Form Consider a non-vertical line L

Locate y-intercept

Let (x, y) be any other point on the line

Find slope m

y

x

L

Lines and Equations

(0, b)

(x, y)

∆x = x – 0

∆y = y – b


Slope-Intercept Form

Lines and Equations

y

x

L

(0, b)

(x, y)

∆x = x – 0

∆y = y – b

Slope-Intercept Form

m(x – 0) = y – b

y = mx + b

Solving for y,

m =∆y∆x

=y – b x – 0


Examples

Point-Slope Form

A line through (5, 10) with slope

Find the equation: y

x

45

(5,10)

m = 4/5=

y – 10 x – 5

Question: Is this form unique ?

m =45


Examples Slope-Intercept Form

A line with intercept (0, 4) and slope -3

Find the equation:

y = mx + b = -3x + 4 y

x

(0, 4)

m = -3Question: Is this form unique ?


Lines and Equations Standard Form

Algebraic form not directly graph related

Useful for systems of linear equations

For constants A, B and C, with B ≠ 0

Ax + By = C

Can we still find slope and intercepts ?

Question: Is this form unique ?


Lines and Equations Standard Form – Slope

Rewriting the equation:

This has slope-intercept form with

= mx + b

y = xAB

– + CB

= mSlope = AB

–

Note:

a ≠ A b ≠ B,


Standard Form Rewriting the equation:

Lines and Equations

= mx + by = xAB

– + CB

Note:b ≠ B= m A

B–

So …

CBb =

= bm– –

CBAB

–

( )( )

= CA

– Intercepts

Fractions


Standard Form Rewriting the equation:

Intercepts

Lines and Equations

y = xAB

– + CB

Note:a ≠ A

y-intercept = (0, b) ( )0,CB=

x-intercept ( )= bm– , 0 = ( )C

A, 0

– Intercepts

b ≠ B

(a, 0)=


Intercepts with Standard Form Consider equation

6x + 5y = 30

When x = 0 ,

5y = 30

y = 6

Vertical intercept is (0, 6)

Standard Form Example

x

y

(0, 6)


Intercepts with Intercept Form Consider equation

6x + 5y = 30

When y = 0,

6x = 30

x = 5

Horizontal intercept is (5, 0)


x

y

(0, 6)

(5, 0)


Intercept Form Consider standard form equation

Ax + By = C

For C ≠ 0 , this becomes

Lines and Equations

AC

x +BC

y = 1

OR intercept form

= 1( )C

A

x+

y

( )CB

Fractions


Intercept Form Consider intercept form equation

Lines and Equations

x

y

( )CA

x yCB( )

+ = 1

0, CB)(

, 0CA( )Question:

What if C = 0 ?

= 1ax

+ by

(a, )0=

( )0, b=

Note:A ≠ aB ≠ b


Finding Intercepts with Intercept Form Consider equation

6x + 5y = 30

Vertical intercept is (0, 6) Horizontal intercept is (5, 0)


x

y

(0, 6)

(5, 0)

301

6x + 5y( )301

30( )=

x5

y6

+ = 1


Horizontal and Vertical Lines Horizontal Lines

Form: y = k

for some constant k

From standard form Ax + By = C when A = 0, B ≠ 0

Question: What is the line y = 0 called ?

CB

y =

The x-axis !


Horizontal and Vertical Lines Horizontal Lines

Example: y = 3 Pick any points

(x1, 3) , (x2, 3) Slope m is then

Note:

Zero slope is NOT the same as no slope

x

y

y = 3 (x1, 3) (x2, 3)

x1 x2

x = x2 – x1

y = 0

m =yx

3 – 3x2 – x1

= 0=


Horizontal and Vertical Lines Vertical Lines

Form: x = k for some constant k

From standard form Ax + By = C when A ≠ 0, B = 0

Then Ax = C

A

Cx =

Question:

What is the line x = 0 ? The y-axis !


x

y Vertical Line Example: x = 4 Pick any points

(4, y1) , (4, y2) Slope m is then

Horizontal and Vertical Lines

x = 4

(4, y1)

(4, y2)

y1

x = 0 m =

yx = 4 – 4

y2 – y1

= 0y2 – y1

y2 y

Undefined !

Note:

No slope is not the same as zero slope !


Parallel Lines Parallel Lines

Horizontal Lines Zero slope, always parallel

Vertical Lines No slope, always parallel

Other lines Lines with same slope, always parallel


x

y

Parallel Lines Parallel Lines Example

Find the equation of the line through

(4, 10) parallel to the line

Slope of new line is

Point-slope form is(4, 10)

y = – (½)x + 6OR

y = – (½)x + 12

12

–

y 12

–= x + 6

=y – 10 x – 4

12

–

= y – 10 (x – 4) 12

–


Step 1 Geometry gives us

Perpendicular Lines in General

x

y

L2

L1

m2

m1

a

c1

c2

c12 = a2 + b1

2

c22 = a2 + b2

2

m1 = b1 a

m2 = – b2

a ,

b1 am1=

b2 – am2=

b1 = am1

b2 = – am2

,

c12 + c2

2 =

b12 + 2a2 + b2

2


Step 2

Perpendicular Lines in General

x

y

L2

L1

m2

m1

a

c1

c2

b1

b2

= – 2a2m1m2

c12 + c2

2 = (b1 + b2)2

b1 + b2 = b12 + 2b1b2 + b2

2

c12 + c2

2

= b12 + b2

2 + 2a2

2a2

= 2(am1)(– am2)

1 = – m1m2

2b1b2 =

– 1m2

m1 =


Perpendicular Lines Example

Example

Find the equation of the line through (2, 3) perpendicular to line

y = –(⅓)x + 3

Slope of the given line is

m1 = –⅓ Slope of the new line is

x

y

(2, 3)y = –(⅓)x + 3

=

3 1 m1

–m2

=

= – 1 ⅓

–


Perpendicular Lines Example

ExampleSlope of the new line is

y =

3x

– 3

= =

3 m2

1 m1

–=

– 1 ⅓

–

Alternate point-slope form

y – y1 = m(x – x1)

y – 3 = 3(x – 2)

Slope-intercept formy = mx – mx1 + y1

y = 3x – 3

x

y

(2, 3)y = –(⅓)x + 3


Variation Direct Variation

A variable y varies directly as variable x if

y = kx

for some constant k

The constant k is called the constant of variation

K is also known as the constant of proportionality


Variation

Direct Variation Example State sales tax t varies directly as

the amount of sale s , i.e. t = ks For tax of $200 on a $12.50 sale,

what is the constant of variation ?

s

tk = ts

=12.50200.00

.0625= k = .0625

Question: Does this look like y = mx + b ?


Inverse Variation Variable y varies inversely as variable x if

for constant of variation k

k is also known as the

constant of inverse

proportionality

Variation

x

y

y = kx


Variation

Inverse Variation Example

At constant temperature the pressure P of a gas in a balloon is inversely proportional to its volume V so that

V

P

P =kV


Think about it !

Documents

Linear Equations Equations of Lines. 7/2/2013 Linear Equations 2 Lines and Equations Point-Slope Form Given line L and point (x 1, y 1 ) on L Let (x,