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