Key Focus #6 - Finding the Slope of a Line Between Two Points.

Given the following equations of lines, find the SLOPES of the lines:

1. 2. 3.

4. 5. 6.

7. 8. 9.

2. Given the following pairs of points, find the slope AND equation of the line.

10. 11. 12.

y = 3x + 6

3x + 8y = 7 9x - 5y = 7 - 5x - 9y = 3

2x = 9y - 8 - x = y + 7 23y = 4x - 19

( 1 , 4 ) and ( - 3 , 5 ) ( - 7 , - 5 ) and ( - 2 , - 1 ) ( 5 , 6 ) and ( - 1 , - 1 )

Answekey1. m = 3 2. m = - 2 3. m = 5 4. m = - 3 5. m = 9 6. m = - 5 7. m = 2 8. m = -1 3 9 8 5 9 99. m = 4 10. m = - 1 x + 4y - 17 = 0 11. m = 4 4x - 5y + 3 = 0 12. m = 7 7x - 6y + 1 = 0 23 4 5 6

Key Focus #6 - Finding the Slope and EQUATION OF A LINE

Find the Slope of the Following Lines:

1. 2. 3.

4. 5. 6.

If each of the following lines are Direct Variations, provide the Equation of each line in standard form, given the “m”:

7. 8. 9. 10.

Find the equation of the line through the given point having the given slope:11. m = 2 , ( - 2 , 4 ) 12. m = - 4 , ( 5 , 6 ) 3 7

13. m = - 1 , ( - 1 , - 2 ) 14. m = 3 , ( 3 , - 5 ) 2

2x + 3y = 11 5x + 8y = - 3 x - 7y = 9

17x - 41y = - 23 - 4x + 2y = 10 - 6x - 3y = 9

m = 2 3

m = - 4 7

m = 1 2

m = 3

Answerkey

1- 2

2.- 5

3.1

417

5. 2 6. - 27. 2x - 3y = 0 8. 4x + 7y = 0 9. x - 2y = 0 10. 3x - y = 038741

11. 2x - 3y + 16 = 0 12. 4x + 7y - 62 = 0 13. x + 2y + 5 = 0 14. 3x - y - 14 = 0

Key Focus # 6 - Points of Intersection Graphing line equations that apply to a real world problem. For each of the five scenarios, write the line equations, graph both lines and state the P.O.I.

Scenario 1:The Acme Yearbook Company charges $300 to set up the layout of the book and $10 for each yearbook.

Equation : ________________________________________

The Quality Yearbook Company charges $100 to set up the layout of the book and $15 for each yearbook.

Equation : ________________________________________

POI : _________________________

When would you use Acme Yearbook Company and when would you use Quality Yearbook Company?

Scenario 2:

The school basketball team is traveling by bus to a competition. The Orange Bus Company charges a flat rate of $250 plus an additional charge of $4 for each athlete that travels on the bus.

Equation : ________________________________________

The Yellow Bus Company charges a flat rate of $150, plus an additional charge of $8.

Equation : ________________________________________

POI : _________________________

Scenario 1: Acme: y = 10x + 30 Quality: y = 15x + 100 POI (40,700) Quality 0 - 40 books Acme >40Scenario 2: Orange: y = 4x + 250 Yellow: y = 8x + 150 POI (25, 350) Yellow 0 - 25 Orange >25

Key Focus # 6 Solving Systems of Equations Scenario 3:The three window cleaning companies offer the following prices:

Price A: $30 for supplies, plus $4 per window Price B: $7 per window. You have 15 windows to clean.

Equation for Price A : ________________________________________

Equation for Price B : ________________________________________

Which would you choose and why?

Scenario 4:Three book binding companies offer the following prices: Price A: $150 for shipping, plus $4.50 per book Price B: $300 for shipping, plus $3.00 per book Price C: $8 per book.

You want to have a maximum of 100 books bound.

Equation for Price A : ________________________________________

Equation for Price B : ________________________________________

Equation for Price C : ________________________________________

Scenario 5: The price for Decorator A is $100 for the first visit, plus $35/h The price for Decorator B is $300 for the first visit, plus $20/h The price for decorator C is $200 for the first visit, plus $25/h

Equation for Price A : ________________________________________

Equation for Price B : ________________________________________

Equation for Price C : ________________________________________

Scenario 3: A: y = 4x + 30 B: y = 7x POI( 10, 70) Use price A for 15 windows Scenario 4: A: y = 4.50x + 150 B: y = 3x + 300 C: y = 8x For 100 books, use CScenario 5: A: y = 35x + 100 B: y = 20x + 300 C: y = 25x + 200 0 - 10, use A, 10-20 use C, >20 use B

Key Focus # 7 1. A square has vertices A( 0 , 4 ) , B ( - 6 , 0 ), C ( - 2 , - 6 ) and D( 4 , - 2 ). Draw the SQUARE on a grid. a) Determine the equations of the lines containing the four sides. b) Determine the equations of the lines containing the two diagonals.

Answerkey AB: BC: CD: AD:

AC: y = 5x + 4 BD:

Key Focus # 7 - Where do they Meet?

Where do they Meet?The Nguyen family lives in Thunder Bay. The Berg family lives in Kenora. At 12 noon, the Nguyens leave by car on a trip to Kenora and drive at an average speed of 70km/h. At the same time, the Bergs leaves by car on a trip to Thunder Bay. They drive at an average speed of 80km/h. It is approximately 510 m from Kenora to Thunder Bay. Where are the two cars when they meet?

1 a)Complete the table to show how far the Nguyen family is from Thunder Bay after each hour of the trip.

Time ( h ) 0 1 2 3 4 5 6 7 8

Distance from Thunder Bay (km)

0

b) Draw a graph of distance from Thunder Bay against time for the Nguyen family

2a) Copy and complete the table to show how far the Berg family is from Thunder Bay after each hour of the trip:

Time ( h ) 0 1 2 3 4 5 6 7 8

Distance from Thunder Bay (km)

510

b) Use the same grid as in exercise 1b. Draw a graph of distance from Thunder Bay against time for the Berg family.

3. The two lines you drew in exercise 1 and 2 should intersect. What are the coordinates of the point where they lines intersect? What do these coordinates tell you about the locations of the two cars?

They meet at 3.4hours into the trip, 238km from Thunder Bay

Key Focus # 7 - Equations of Lines

Complete the Chart

Equation Slope y-intercept Slope of Line parallel to this line

Slope of Line Perpendicular to this Line

1. y = 5x + 1

2. y = - 2x - 7

3. y = 3x + 2 5

4. 2 7

5. 8 - 4

6.3

1 4

7.5 - 2

8.0

2 3

9. y = x

10. y = 7

11. x = 2

2. a) On a grid, draw a line with slope 2 through A ( - 3 ,2 ) 5 b) Draw a line through C ( - 2 , - 1 ) that is parallel to the line in part a. c) Draw a line through C ( - 2 , - 1 ) that is perpendicular to the line in part a.

3. a) On the same axes, draw a line through P( 5 , 2 ) with each of the following slopes: i) 2 ii) - 1 iii) - 5 iv) 0 v) Not defined by a real number. 3 4 7 b) Choose one slope from part a. Write to explain how you drew the line.

4. Which of these points lie on the line y = 2 x + 4 5 5 a) C ( 3 , 2 ) b) D ( 7 , 4 ) c) E ( - 5 , 1 ) d) F( 8 , 4 ) e) G ( 0.5 , 1)

5. Determine the equation of the line that passes through each pair of points. a) A ( 3 , 5 ) and B ( - 5 , - 3 ) b) C (- 4 , 7 ) and D ( 5 , - 4 )

6. Determine the equation of the line that passes through J ( 5 , 5 ) and is : a) Parallel to the line y = - 3 x - 4 b) Perpendicular to the line y = - 5 x + 5 4 2

7. Graph each line without making a table of values. a) 2x - 3y + 12 = 0 b) 3x + 5y = 0 c) 5x + 2y - 15 = 0

8. Solve each linear system. a) x + y = - 8 b) x + 2y = 6 x - 2y = 7 2x - 3y = 5

Key Focus # 7 - Answers

1. Complete the Chart 2. Graph

3. 4. a) Yes b) No c) No d) Yes e) Yes

5. a) y = x + 2 b)y = - 11 x + 19 9 9

6. a) y = - 3 x + 35 b) y = 2 x + 3 4 4 5

7. Graph 8. a) ( - 3 , - 5 ) b) ( 4 , 1 )

Equation Slope b parallel Perpendicular

1. y = 5x + 15 1 5

- 1 5

2. y = - 2x - 7- 2 - 7 - 2

1 2

3. y = 3x + 2 5

3 5

2 3 5

- 5 3

4. y = 2x + 72 7 2

- 1 2

5. y = - 4x + 8- 4 8 - 4

1 4

6. y = -4x + 3- 4 3 - 4

1 4

7. y = 1 x + 5 2

1 2

5 1 2

- 2

8. y = 2 x 3

2 3

0 2 3

- 3 2

9. y = x1 0 1 -1

10. y = 70 7 0 undefined

11. x = 2undef

doesn’t cross

unde-fined

0

i) y = 2x - 4 3 3

ii) y = -1x + 13 4 4

iii) y = -5x + 39 7 7

iv)y = 2

v) x = 5

c) y = -5x + 15 2 2

a) y = 2x + 4 3

b) y = - 3x 5

Grade 9 Assessment of Mathematics ACADEMICStudent Answer SheetYour multiple-choice answers must be entered on this sheet• To indicate your answer, use a pencil to fill in the circle completely

Like this: Not Like This:• Do not fill in more than one answer to a question• Do not leave a question blank.• Cleanly erase any answer you wish to change and fill in the circle for your new answer

x

1. a b c d

2, a b c d

3. a b c d

4. a b c d

5. a b c d

1. Which of the following points is in the third quadrant of the xy-plane?

a) ( 5 , 7 )b) ( 5 , - 7 )c) ( - 5 , 7 ) d) ( - 5 , - 7 )

2. The points ( - 3 , - 4 ) and ( 6 , 2 ) are marked on an xy-plane.

Which statement about the line through (–3, –4) and (6, 2) is not true?

a) Its x-intercept is 3. b) Its slope is positive. c) Its y-intercept is –2.d) It passes through (4, 9).

3. Which graph is the best match to a sketch of y = –3x – 4?

3. Which graph is the best match to asketch of y = –3x – 4?

A

B

C

D

4. Nicole rides her bike to school in themorning. She stops at a store for about5 min when she is halfway to school.Which graph below best describes thissituation?

F

G

H

J

0

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Time

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Dis

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Time

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Dis

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avel

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Dis

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Time0

y

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y

x0

y

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G r a d e 9 A s s e s s m e n t o f M a t h e m a t i c s2

3. Which graph is the best match to asketch of y = –3x – 4?

A

B

C

D

4. Nicole rides her bike to school in themorning. She stops at a store for about5 min when she is halfway to school.Which graph below best describes thissituation?

F

G

H

J

0

Dis

tanc

e tr

avel

led

Time

0

Dis

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avel

led

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G r a d e 9 A s s e s s m e n t o f M a t h e m a t i c s2

16. a b c d

17. a b c d

18. a b c d

19. a b c d

20. a b c d

11. a b c d

12. a b c d

13. a b c d

14. a b c d

15. a b c d

6. a b c d

7. a b c d

8. a b c d

9. a b c d

10. a b c d

4. PQ and RS are parallel line segments. What is the value of y?

a) 5 b) 6 c) 7 d) 8

5. If A is (3, 4) and B is (7, 12), which point is on the line segment AB?

a) ( 3 , 5 ) b) ( 4 , 8 ) c) ( 5 , 9 )d) ( 6 , 10 )

6. A is the point ( - 2, 1), B is the point (1, - 4) and D is the point (1, 6).

If ABCD is a rhombus, which of the following is point C? a) ( 1 , 1 ) b) ( 1 , 4 ) c) ( 4 , 1 ) d) ( 4 , 4 )

7. Which of the following tables of values represents the linear relationship y = 2x?

8. Which of the following properties does the line have?

a) positive slope and positive y-intercept

b) positive slope and negative y-intercept

c) negative slope and positive y-intercept

d) negative slope and negative y-intercept

a) x y0 12 45 25

b) x y0 13 84 16

c) x y0 01 23 6

d) x y0 02 46 36

9. Which points on the graph have the same y-coordinates?

a) A and B b) A and C c) B and C d) B and D

10. The graph shows a scatter plot and a line of best fit.

Which of the following statements about the line of best fit is true?

a) The line has a negative slope and a negative y-intercept.

b) The line has a negative slope and a positive y-intercept.

c) The line has a positive slope and a negative y-intercept.

d) The line has a positive slope and a positive y-intercept.

11. Four points, A, B, C and D, are marked on an xy-plane and joined by line segments as shown.

Which line segment has a negative slope?a) BA b) BC c) CD d) AD

12. Which of the following equations is equivalent to 3x - 5y = 45?

a) y = 3 x - 9 5b) y = - 3 x + 9 5c) y = 3x - 45

d) y = - 3x + 45

13. Which of the following lines has the same slope as the line 2x - y + 1 = 0 a) y = - x + 1 b) y = - x + 2 c) y = - 2x - 1 d) y = 2x - 2

14. Which graph below is likely the graph for y = 2x?

a) b)

c) d)

15. The total cost, C, in dollars, of running an advertisement in a newspaper is made up of an initial cost of $12, plus a charge of $5 per day, where n represents the number of days.

Which equation represents this relationship? a) C = 12n + 5 b) C = 12 + 5n c) C = ( 12 + 5 )n d) C = 12 + 5 ÷ n

17. Natasha works for a computer company. The table shows her annual salary in the last five years.

If the trend continues, what will Natasha’s annual salary be in the 8th year?

a) $40 000 b) $43 200 c) $46 400 d) $49 600

18. Josie works in a sports store. She receives 8% of the total sales each day. One day, she receives $35 for her portion of the total sales. What are the total sales for that day? a) $37.80 b) $43.75 c) $280.00 d) $437.50

16. Ivan sells programs at a baseball game. He earns $20 per game plus $12 for each box of programs he sells.

If x represents the number of boxes of programs he sells at the game, which of the following expressions represents Ivan’s earnings, in dollars, at that game?

a) 20 + 12x b) 12 + 20x c) 12(x + 20) d) 20(x + 12)

19. How would the graph of the relation y = 3x - 2 change if the 3 and - 2 were both doubled?

The graph would be: a) steeper and have a lower y - intercept. b) steeper and have a higher y-intercept. c) less steep and have a lower y-intercept. d) less steep and have a higher y-intercept.

3. The points (–3, –4) and (6, 2) aremarked on an xy-plane.

Which statement about the linethrough (–3, –4) and (6, 2) is not true?

A Its x-intercept is 3.

B Its slope is positive.

C Its y-intercept is –2.

D It passes through (4, 9).

4. Which graph below is likely the graphfor y = 2x?

y

x0

y

x0

y

x0

y

x0

F

G

H

J

* G1(1)

y

x4

1

5 6 7 8 9–4 –3 –2 –1 1

2

3

4

5

6

7

–7

–6

–5

–4

–3

–2

–10 2 3

M a t h e m a t i c s B o o k l e t 1 , A c a d e m i c P r o g r a m 3

3. The points (–3, –4) and (6, 2) aremarked on an xy-plane.

Which statement about the linethrough (–3, –4) and (6, 2) is not true?

A Its x-intercept is 3.

B Its slope is positive.

C Its y-intercept is –2.

D It passes through (4, 9).

4. Which graph below is likely the graphfor y = 2x?

y

x0

y

x0

y

x0

y

x0

F

G

H

J

* G1(1)

y

x4

1

5 6 7 8 9–4 –3 –2 –1 1

2

3

4

5

6

7

–7

–6

–5

–4

–3

–2

–10 2 3

M a t h e m a t i c s B o o k l e t 1 , A c a d e m i c P r o g r a m 3

Year Annual Salary ($)

1 32 000

2 33 600

3 35 200

4 36 800

5 38 400

20.Two bicycle repair stores charge an initial fee and an hourly rate for repairs.

The graph below shows the total cost of repairs versus time for the repair.

Which statement is true?A The two stores charge different hourly rates

and the same initial fee.B The two stores charge the same hourly

rate and different initial fees.C The two stores charge different hourly rates

and different initial fees.D The two stores charge the same hourly

rate and the same initial fee.

Answers:1.D 2.C3.B4.D5.D6.C7.C8.B9.D10.D11.D12.A13.D14.A15.B16.A17.B18.D19.A20.B