NOTES: Writing Equations

A° If an equation is in slope-intercept form, y =mx + b, what two pieces of information do

you know? ÿ\tbÿ ÿ ÿ't%ÿgl Ceÿ(

B° What two pieces of information do you need to know to write an equation

in slope-intercept form? ÿ(5ÿ ÿ, ÿ-\%kggCÿ

C. How do you find these pieces of information from a .............

Graph Table Two points

kU, b

D. What do you do with the slopeequation in slope-intercept form?

and y-intercept when you write the

Write an equation in slope-intercept form.

°

mÿ

°

u99Z

x y-8 45

-6 40

-4 35

-2 30

0 252 20

'L

3. (o, 2)and (7,-5) . x 0 2 4 6 8 10y 7 10 13 16 19 22

°ÿÿ9ÿÿ ÿ lÿÿ°ÿ

.

/€

L

• It

I.I

6. (-5,3) and (0, 9),

Write an equationÿe-intercept form if the slope is 8 and the y-interceptis 12.

8. The point (3, 2) is on the line y = 2x + b. Find the value of b.

tt

9. Drivers in distress near Katy have two towing services to choose from: Brook's Body Shopcharges $3 per mile for the towing, and a fixed $25 charge regardless of the length of the tow.Morgan Motors charges a flat $5 per mile. Write an equation for each situation. If you needto be towed 26 miles, which service would you call and why? If you need to be towed 4 miles,which service would you call and why?

10. Select the equation that represents the graph below.

\\\\

3A. y=- x+3 c. y=--x-3

2

3 3B. y=---x-3 y =---x+3

2 2

11. Identiÿ the slope and the y-intercept from the equation below and ÿ.

9y = 3x + 18

3

The equation Y-Y1 = ITI (X - X1) is in point-slope

!

IformJ

a) What information did you use to set up this problem?

b) Why do you think it is called point-slope form?

(ÿkgeV'ÿ

Write an equation of a line in point-slope form given a point and the slope.

112. (3, 5) m=--

2

13. (-4, 8) m=6

14.5

(-10,-1) m =-6

15. Refer to the information in the table to complete the following steps.hrs cost a. Find the rate of change(slope) from the table.

6 ÿrÿq t.

3 15

b. Write an equation in point-slope formusing the first point in the table.

c. third point in the table. d. fifth pqint in the table.

e) For each of the three equations above distribute the tlÿree and solve for y.

f) What do you notice about the equations? 4ÿ ÿ

,ÿ o,\ÿ ÿ\ÿ 5,ÿb ÿ.'ÿ, '2ÿ¢''ÿ "

g) Does it matter what point you pick to put in point-slope form?

h) What form are the final equations in?

16. Write an equation of a line in slope-intercept form.

a) (-3ÿ 2) and (-:1,-2) !

c) (8, 4)and (4, 3)

ggoY, t :" W'ÿ" tÿ-g t4"ÿ"

17. A jet, cruising at 26,400 feet, begins its d scent into Logan Airport, which is 48 milesaway. Another jet, cruising at 31,680 feet, is 60 miles from Logan when it begins its descent.Write the equation to model each plane's path.

18. At noon my odometer reads 6,852 miles. At 3:30 PM it reads 7034 miles.

a) What is my average rate of change during these 3 and a half hours?

b) Let t represent the number of hours I've been driving since noon and y represent myodometer reading. Write an equation that relates y and t.

c)of this problem.

q \\'t,ÿ, £'2 (,5'ÿ t ÿ ÿ5'2

Show that point (5, 7112) is on your line, and then interpret this point inthe context

\ÿ eÿ ÿ3 x-ÿo,ÿw',

"1, I \ "t.."Z.ÿ ÿ,ÿ'85'L

qÿ \\'t,= qÿ \\'t,19. Avery's long distance phone company charges a connection fee in addition to a per-minute charge for each call. Avery's most recent bill included a $4.24 charge for a 12-minutecall and a $6.00 charge for a 20-minute call.

a) What is the per-minute charge? [ÿ\(ÿ

b) What is the connection fee?

c) How much would Avery be billed for a five-minute call?

7

20. Find an equation in ÿ-slope form for the line containing (-3, 0) and (0, 4).

21. Write an equation in point-slope form fora) the line that goes through (2, 5) and (6, -3).

b) the line that goes througÿpÿoint (h, k) and that has slope m.

22. Casey goes for a bike ride from Brookshire to Katy, while the odometer keeps acumulative record of the number of miles traveled. The equation m = 12t + 37 describes theodometer reading m after t hours of riding. What is the meaning of 12 and 37 in the contextof the trip?

II

23. The table shows the cost of visiting a working ranch for one day and one night fordifferent numbers of people.

# of people 4 6 8 10cost $ 250 350 450 550

a) State two ordered pairs from the table. (ÿ\Cÿ(ÿ

b) Find the slope/rate of change.

c) Write an equation in point-slope form.

d) Then convert to slope-intercept form.

/- 9. 50 = 5ÿo ¢oo4qÿ6 ÿq60

e) Use the equation to find thecost for 60 people.

p\ÿx5 ÿvÿ bÿ ÿOf KI6ÿCe X ÿeÿqesÿkS 1ÿ<bD\g.'I- ÿ(ÿtÿo)ÿS(ÿ.... ,I= ÿ,ÿ ,ÿx:ÿ.... ,iÿ9ÿ,oÿ:ÿ

i) How many people could visit for $950?kÿ reDfeSeÿk %

U6O %o

xÿ \g 9eo9kÿ

i- g J

24.problem for it.

Write an equation for the line that contains the points on the table, and make up a word

XY

9iCÿX\ 0XI\ 100

Xz-ÿ,t

15 ×ÿ 30160 \ lÿ 220

45280

60340

.... q t

25. Answer the questions below to GRAPH the line given in point-slope form.

y + 3 = ÿ (x + 5)

at Name the point. # ÿ ÿ° MO,\kÿ

b) What is the slope? * / ÿ ....

Plot the point on the graph.c)

d) Then use the slope to determinetwo more points on the line.

[\ÿ5)

2h A?ÿ.2d5s " last phone bill he wasar (m,c) to write an equation

in point-slope form.

Uÿ-ÿ ÿ %.ÿ5

xI°',Iÿ: wÿ (x.- ÿ6

27. If (-3, 5) and (2, 1) are two points on a line, find three other points on the same line.

i0

Standard form of a line: Ax+By =C

® A, B, and C are always integers.

® A is always positive.

28. Rewrite the equation 2y =-4x-3 in standard form ax+by = c (Cÿeÿ ÿ ÿ. ÿt eva acvraÿ

3-x=4y+2

29. Rewrite the equation 2 in standard form.

30. Rewrite the equation 3y = 4x + 2 in standard form. ÿ¢ÿt

31.

7 11y=--x--

Rewrite the equation 3 8 in standard form.

82. Write the equauon m srancrarct mr of a line with a slope of 3 and passing

through(2,4). ¢ ÿtt%ÿc Obÿ Iÿ DOtÿ{-%\Cÿlÿe ÿ'tSfÿ ÿeeÿtÿrÿ goÿvÿgeÿ

t4

Writing Equations Review for

33.

Quiz:

Write an equation of a line in slope-intercept form that passes through the points

34. Write an equation of a line in slope-intercept form that passes through the

point (8,-3) and has the slope of -5.

Name the slope and one point on the line from each point-slope form equation.

sÿVÿ 1 4)35. y+5 =-3(x-9) 36. y-13=s(x-Y

37. Graph a horizontal line through the point (3, 5). Choose another point on this line.What is the slope of this line? What is the y-intercept of this line? Write an equation for thiline.

38. Write the equation of the line graphed below in slope-interce

(ÿ\ÿ ÿL 5ÿ ÿ)oÿs.ormÿ

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12

- (

\

39. Write an ÿm of the line passing through the given points.

a) (2, 3) and (4, 9) b) (3, -3) and (6, -1)

o D

4

40. The manager at Ben and Jerry s Ice (ÿream Company estimates the cost C (in dollars) ofproducing n quarts of ice cream in a given week is given by the equation C = 550 + 1.20n.

a)were made that week?

\ ÿ ÿ \.qÿ IJ

During one week, the total cost of making ice cream was $1070. How many quarts

\,t \'%

b) Explain the meanings of"560" and "1.20" in the cost equation.

41. When weights are placed at the end of a spring, the spring stretches. If a three-poundweight stretches the spring to a length of 4.25 inches, a five-pound weight stretches thespring to a length of 5.75 inches, and a nine-pound weight stretches the spring to a length of8.75 inches. What is the initial weight of the spring?

(ÿÿ '.

13

After Thanksgiving Review:State the form and describe how we graph from each form:

Slope-Intercept Form:I

Point-Slope Form: Standard Form:

{xÿ- <ÿ ÿ ¢ÿ

%

42.

(9, -7) and (6, -5).a) Write an equation in point-slope form for a line that passes through the points:

b) Rewrite your equation into slope-intercept form:

'V-ÿ=l= °?ÿ,/.4 ÿ

c) Rewrite your equation into standard form.

,%<

14

43. Given the points A(1,-2), B(0,-5) and C(-1,-1), D(0,2).a. Find the slope of the line containing the points A and B.

b. Find the slope of the line containing the points C and D.

e. Write the equation of the line passing through the points A and B.

d. ÿuation of the line passing through the points C and D.

e. What are the similarities and differences in these two equations?

f. Make a prediction, based on the slopes of the lines, as to what kind of lines these will

be. What is your reasoning for making this prediction? (Be specific)

g. Graph line AB and CD.!

/ //

¢ \ l ÿ"

h. What kind of lines are these?

i, Explain how you would describe _ÿÿ lines based on their slopes.

15

44. Graph y = 2x- 4

Graph a line with a slope of 2that goes through the point (-3, 4)

What do you notice about the linesand their slopes?

---j

"!'

/ !lJf

J7L-¥

***These lines are ÿCÿ\ÿ\

because they have the same ÿ\ÿ ÿ

45. Write the equation of a linethat is parallel to the graph ofy = 3x - 2 and contains the

point (2, -3).

46. Write the equation of a line thatis parallel to the graph of 2x + 4y = 8and contains the point (-5, -15).

Scatter PlotsA scatter plot is a graph used to determine whether there is a relationship between paireddata. Scatter plots can show trends or patterns in data.What trend do you notice in each of the scatter plots below?

16

47. BIRD POPULATIONS The table shows the number of active red-cockeded woodpeckerclusters in a part of the De Soto National Forest in Mississippi. Write an equation thatmodels the number of active clusters as a function of the number of years since 1990.

Year 1992 1993 1994 1995 t996 1997 1998 1999 2000

Active 22 24 27 27- 34 40 42 45 51Clusters

joÿo

q

Q

6

@

o

5

%

b ÿtotÿ

b

Clusters in 1997 ÿ(ÿ

Clusters in 2010 ÿÿ

When are there 38 clusters? %ÿÿ \ÿb ÿ, \ÿ7

When are there 59 clusters?

%,b %.b

48. Make a scatter plot of the data anddescribe the correlation of the data. Ifpossible, fit a line to the data andwrite the equation of that line.

X 10 12 15 20 30 45 60 99Y -2 4 9 16 32 55 87 128

v V N] V v

49. Make a scatter plot of the data anddescribe the correlation of the data. Ifpossible, fit a line to the data andwrite the equation of that line.

bO

//

J/

J/

/¢

#

/

\tO

\

i

v

X -5 -3 -3 0 1 2 5 6Y -4 12 10 -6 8 0 3 -9

A(b Obÿ ÿ ÿkÿtÿorÿ

18

50. Your school's sailing club wants to buya sailboat. The table shows thelengths and costs of sailboats.

A A A ^ A ALength 11 12 14 14 16 22 23(ft)Cost 600 500 1900 1700 3500 6500 6000(dollars) ¢\ÿ \ÿkÿ "ÿ,ÿ \ÿ ÿ

t

gOoO

\

4@'/.--qqO6 \ÿ.ÿ.ÿ, Q63ÿ't

: %66

-9c _506\

\t \ÿ to tÿ\

Make a scatter plot. Let x represent the length and let y represent thecost.

b. Find an equation that models the cost (in dollars) of a sailboat as afunction of it's length (in feet).

c. Approximate the cost of a sailboat that is 20 feet long.

d. If a sailboat cost $5000, approximately how long is it?

19