Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line...

39
Linear Equations

Transcript of Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line...

Page 1: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Linear Equations

Page 2: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Linear Equations

• Graph forms a line

• To identify whether an equation is linear, the two variables are to the 1st power, +, -, x, / of a variable by a constant

• Ex. of linear eq Ex. of non-linear

2

336

5

432

32

xy

ts

x

yx

xy

xy

xyx

xy

ba

1

4

1

1287 2

Page 3: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Standard Form for Linear Equations

34

:form standard toChange

possible if termsallon factor common by reduce

zeroboth not are b and a

fractionsnot are constants and tscoefficien

positive bemust a

x-y

cbyax

Page 4: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

2035

3

:form standard toChange

yx

Page 5: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Slope Intercept form of a line

1243

formintercept slope toChange

intercepty b

line theof slopem

yx

bmxy

Page 6: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Change from standard to slope intercept or vice versa

23

2

3

4

3

123

xy

yx

Page 7: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Graphing a line—using x and y intercepts

042

line graph the andequation following

for the interceptsy and x theFind

0xintercept y at the

0yintercept x at the

axisy thecrossesgraph the whereisintercept y

axis x thecrossesgraph the whereisintercept x

yx

Page 8: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Solutions to an equation

• The solution to a linear equation is the ordered pair (x,y) that makes the equation true

• The graph of a line represents all of the points that would be solutions to the equation

Page 9: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Determine whether (-1,1) is a solution?

32 yx

Page 10: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Graphing lines

• You can graph a line as long as you have two points on the line

• You can also graph from a x/y chart by plotting the points and drawing the line

• You can also graph from slope intercept form of a linear equation

Page 11: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Graphing a line using slope intercept form

• m=slope (steepness and direction of the line)

• b=y intercept

• Positive slope Negative slope

• Zero slope Undefined slope

Page 12: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Equations of lines

• If you have a zero slope, your equation will be y = #

• If you have a line with an undefined slope, your equation will be x = #

Page 13: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Finding Slope

• To find slope (rate of change): – From 2 points—use point-slope formula

– From a graph—use rise/run

– From an equation—put in slope intercept form

– Slope formula:

12

12

xx

yym

Page 14: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Find the slope given the following 2 points

)2,5()2,5(

)4,5

2()

3

1,3(

3

2,

4

1,

2

14

)2,4()1,4(

)5,2()4,3(

and

and

randm

and

and

Page 15: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Find the slope from the graph

Page 16: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Graphing from slope intercept form

• Plot any point on the line (can be the y intercept) and from that point use the slope to continue graphing

223 yx

Page 17: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Graphing from slope intercept form

84 yx

Page 18: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Graph the line

63 x

Page 19: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Graph the line

132 y

Page 20: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Graph the line

xy

Page 21: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Graph the line

3

1)3,2( mthru

Page 22: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Graph the line

undefinedmthru )2,1(

Page 23: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Graph the line

0)2,5( mthru

Page 24: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Graph the line )2,2(1 thrugoesandxtolarperpendicu

Page 25: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Writing Equations of Lines

• To write the equation of a line, you must know a point and the slope

• Use the point slope formula to write the equation

• Put the final equation in slope intercept form

.),(

)(

11

11

pttheisyx

slopem

xxmyy

Page 26: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Write the equation of the line

• From a point and the slope

7,5

3

1

m

3,1

4

3

m

2,4

undefinedm

Page 27: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Write the equation of the line

• From 2 points—must find the slope first

4,3,3,6 3,1),3,4(

2

1,

4

1,

2

1,

4

3

Page 28: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Parallel lines

• Parallel lines have the same slope

• To write the equation of a line given the parallel line, use the same slope

23

)3,3(

yxtoparallel

Page 29: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Parallel lines

• To write the equation of a line given the parallel line, use the same slope

84

)6,3(

xtoparallel

Page 30: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Perpendicular lines

• Perpendicular lines have opposite reciprocal slopes

• To write the equation of a line given the perpendicular line, flip the slope and change the sign

652

)4,1(

yxtolarperpendicu

Page 31: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Perpendicular lines

• To write the equation of a line given the perpendicular line, flip the slope and change the sign

93

)7,7(

ytolarperpendicu

Page 32: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Perpendicular lines

• To write the equation of a line given the perpendicular line, flip the slope and change the sign

xytolarperpendicu

)5,9(

Page 33: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Write the equation of a line from a graph

Page 34: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Scatter Plots

• Used to show association or relationship between data—can have strong or weak association

• Line of best fit (trend line or line of regression)—can be used to make predictions

Page 35: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Scatter Plots

Positive Association Negative Association No Association

Independent Variable—x

Dependent Variable--y

Page 36: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Create a Scatter Plot

Year Population

2000 122,900

2002 121,500

2004 130,000

2006 140,000

2008 152,500

2010 169,000

2020 ?

Determine the equation for the

regression line?

What would be the population in

2002?

Page 37: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Create a Scatter Plot

Year Avg cost of

home

2000 156,000

2001 189,100

2002 201,300

2003 204,800

2004 212,300

2005 220,550

2006 237,400

Determine the line of best fit and

its equation

What would be the avg. cost of a

home in 2010?

Page 38: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Shoe Size and Height Name Height (in

inches)

Shoe Size

Page 39: Linear Equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •To write the equation of a line given the parallel line, use the same slope 4 8 (3 , 6 ) parallel to x. ... slopes

Shoe Size and Age

Name Height Shoe Size

Ms. Jones 67 inches

Based on the data recorded regarding your height and your shoe

size, what type of association exist?

Make a prediction for my shoe size if my height is 67 inches?

Does this seem reasonable?