linear equations · 1 4 ( 4 ,1) ( 4 ,2) ( 3 ,4 ) ( 2,5 ) ... •to write the equation of a line...

Linear Equations

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

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

2035

3

:form standard toChange

yx

Slope Intercept form of a line

1243

formintercept slope toChange

intercepty b

line theof slopem

yx

bmxy

Change from standard to slope intercept or vice versa

23

2

3

4

3

123

xy

yx

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

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

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

32 yx

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

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

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 = #

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

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

Find the slope from the graph

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

Graphing from slope intercept form

84 yx

Graph the line

63 x

Graph the line

132 y

Graph the line

xy

Graph the line

3

1)3,2( mthru

Graph the line

undefinedmthru )2,1(

Graph the line

0)2,5( mthru

Graph the line )2,2(1 thrugoesandxtolarperpendicu

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

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

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

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

Parallel lines

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

84

)6,3(

xtoparallel

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

Perpendicular lines

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

93

)7,7(

ytolarperpendicu

Perpendicular lines

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

xytolarperpendicu

)5,9(

Write the equation of a line from a graph

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

Scatter Plots

Positive Association Negative Association No Association

Independent Variable—x

Dependent Variable--y

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?

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?

Shoe Size and Height Name Height (in

inches)

Shoe Size

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?