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Unit 5Two Variable Linear Relations &

Solving Linear Equations

Learning Targets:

#1: I can represent patterns using linear equations, tables of values, graphs and words.#2: I can interpret linear graphs using interpolation and extrapolation.#3: I can solve one-step equations with rational numbers.#4: I can solve two-step equations with rational numbers.#5: I can solve equations including distributive property.#6: I can solve equations with variables on both sides.#7: I can solve word problems algebraically.

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Lesson 1 – Representing Patterns

A linear relation is a _______________________ between two sets of _______________________ that give a _______________________ when graphed.

1. a) Draw the next two figures in this series.

b) Create a table of values comparing the number of squares and the figure number.

Figure Number, x Number of Squares, y

c) Describe the pattern.When the figure number ___________________________________, thenumber of squares ___________________________________.

d) Write the equation that represents this pattern.

e) How many squares are in Figure 15?

f) Which figure number has 69 squares?

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2. What linear equations model the relationship between the values in each table?

3. A number pattern starts at 1.5. Each number after that is four more than the number before.a) Make a table of values for the first five terms.

Term Number, x Number, y

b) Develop an equation that can be used to determine the value of each term in the pattern.

c) What is the value of the 95th term?

d) Which term has a value of 237.5?

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a) x 0 1 2 3

y 11 16 21 26

b) x  1  2 3 4

y –2.1 –0.6 0.9 2.4

4. On top of the $45 monthly fee, Sam’s cell phone plan charges $0.15 for every text message he sends or receives.a) Develop an equation to calculate the monthly bill.

b) What would Sam’s bill be if there were 20 text messages in a month?

c) If Sam budgets $80 a month for his cell phone, how many text messages can he send or receive each month? Explain.

Practice:118 example: t = time

1. Choose a variable to represent each unknown.

a) distance to stop b) row number

c) cost of a ticket d) age of a person

2. Rewrite the statement as an algebraic equation.

The cost of a monthly bus pass is 20 times the price of a day ticket

c = cost of a pass; t = price of a ticket

equation: ____________ = ____________

3. For each table, describe a pattern to go from the input to the output. Then, write an equation for each.

a. b.

For the pattern, describe the change in x and the change in y as you move down each list of values.

Describe the pattern: Describe the pattern:

Write the equation: Write the equation:

4. a) Complete the table of values.

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Ask yourself, “Do I add, subtract, multiply, or divide?”

Input, x Output, y

3 301 104 408 80

Input, x Output, y

1  73  95 116 12

b) Give the linear equation to find the number of circles in each figure.

5. A rectangular table seats 6 people:

You connect the tables end to end. Each time you add a table, another 4 people can sit down:

a) Complete the table of values.

Number of Tables, x

Number of People, y

1 6234

b) Write an equation.

c) How many connected tables will seat 26 people?

6. Laura uses black tiles and white tiles to create a pattern.

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FigureNumber, x

Number of Circles, y

a) Complete the table of values.

FigureNumber, x

Number of Black Tiles, y

b) Write an equation to find the number of black tiles in each figure.

c) How many black tiles are in Figure 24?

d) Which figure has 176 black tiles?

7. Jessica’s number pattern starts with –5. Each following number is 3 less than the number before it.

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a) Complete the table of values for the first 5 terms in the number pattern.

Term Number, x Term Value, y

1 –52 –5 – 3 = 345

b) Using the table of values, determine the equation.

c) Use the equation to find the value of the 20th term.

d) Use the equation to find the term that has a value of –119.

8. Write a linear equation for each table of values.

Equation: Equation:

9. A company charges your school $125 to design new T-shirt logos. It costs an extra $15 to make each T-shirt.

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x y0 131 162 193 22

x y0 171 242 313 38

a) Complete the table of values.

Number of T-shirts Cost ($)  0 0 × 15 + 125 = 125  5 5 × 15 + 125 = 101535

b) Find an equation to model the pattern.

First variable, , represents .

Second variable, , represents .

Equation:

c) What will it cost to make 378 T-shirts?

d) How many T-shirts can the school order for $2345?

10. Figure 1 has 2 regular hexagons connected along 1 side.Each side length measures 1 cm.

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The perimeter of Figure 1 is 10 cm.

Figure 1

a) The next figure has another hexagon connected to it on 1 side.Draw the next 3 figures. Find the perimeter for each.

Figure 2 Perimeter =

Figure 3 Perimeter =

Figure 4 Perimeter =

b) What equation can you use to find the perimeter of each figure?

c) What is the perimeter of Figure 12?

d) How many hexagons do you need to create a figure with a perimeter of 118 cm?

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Count the number of sides in each drawing.

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Lesson 2 – Interpreting Graphs

Interpolate – estimate values _____________________ known data.

Extrapolate – estimate values _____________________ known data.

1. a) What is the approximate value of

d when t = 3? ______

Explain the method you used.

____________________________________________________

____________________________

b) What is the approximate value of

t when d = 300? ______

2. a) What is the approximate value of

y when x = –1.5? ____

b) What is the approximate value of

x when y = 10? ____

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3. a) What is the approximate value of

y when x = 3.5? ____

b) What is the approximate value of

x when y = 0.5? ____

4. a) In the deli section of a grocery store, Greek salad costs $1.50 per 100 g. Plot the data on a graph.

Mass of Greek Salad, m (g) 100 200 300 400 500

Cost, C ($) 1.50 3.00 4.50 6.00 7.50

b) From the graph, determine the cost of 800 g of Greek salad.

c) From the graph, determine how much salad you get for $10.50.

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5. A car rental company charges a flat rate of $35.00 plus $0.45 per kilometre for renting a car. The graph shows the cost of renting a car based on the number of kilometres driven.

a) Is it reasonable to interpolate or extrapolate values on this graph? YES NO Explain._________________________________________________________________

_________________________________________________________________

b) What is the rental cost after driving 300 km?

c) Approximately how many kilometres can be driven for a rental cost of $115?

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Lesson 3 - Graphing Linear Relations

1. Suri drives at an average speed of 90 km/h. The equation relating distance, d, and time, t, is d = 90t.

a) Complete a table of values to represent the relation.

b) Show the relationship on a graph.

c) How long does it take Suri to drive 630 km?

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Time, t Distance, d

2. For each linear equation, create a table of values and a graph. x y

–3

–2

–1

 0

 1

 2

 3

a) y=– 2 x – 4

x y

–3

–2

–1

 0

 1

 2

 3

b) y=– 3

c) y= x4−2

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x y

–3

–2

–1

 0

 1

 2

 3

3. Create a graph and a linear equation to represent each table of values.

a) x y

–3 4

–2 4

–1 4

 0 4

 1 4

 2 4

 3 4

b) x y

5   4

6 4.5

7   5

8 5.5

9 6

10 6.5

c) x y

0 –2.0

1 –1.75

2 –1.5

3 –1.25

4 –1

5 –0.75

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4. The graph shows the relationship between the fuel consumption, f, in litres (L), and the distance driven, d, in kilometres (km).

a) What is the linear equation?

b) How far could you drive with 34 L of gas?

c) Is it appropriate to interpolate or extrapolate values on this graph? What assumption is being made? Explain.

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Practice:

1. a) For the linear relation y = 2x – 5, complete the table of values, graph your points, then join them with a line and extend your line in both directions.

x y

–2

–1

 0

 1

 2

b) Use the graph to estimate the value of y when x = 8. y =

c) Use the graph to estimate the value of x when y = – 4. x =

2. a) Complete the table of values for the graph.

b) The linear equation that represents this graph is .

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t d0 4

3. Ian works part-time at a movie theatre. He earns $8.25/h.

The equation p = 8.25t models the relationship between his pay and hours worked. p = pay; t = time in hours.

a) Complete a table of values for this relation.

Time, t (h) Pay, p ($)

0

1

2

3

4

b) Graph the relation. Label the axes and write the title.

c) Ian works 8 hours in 1 week. Use the equation to find his pay.

p = 8.25t

Substitute t = 8.

p =

p =

d) Use the graph to extrapolate his pay for 8 hours of work:

Is this answer close to the answer for the previous questions? YES NO

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4. For each equation, complete the table of values. Use the table of values to help you match the equation to its corresponding graph.

y = 5x

x y– 4–2 0 2 4

The matching graph is .

y = –2x + 3

x y–2 0 2

The matching graph is .

y = + 6

x y–2 0 2

The matching graph is .

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5. Create a table of values and a graph for each linear equation.

a) y = 4

x y0246

b) y = –3x + 4.5

x y0123

c) y = x5 + 13

x y  –5  0  510

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6. The graph represents a relationship between the water height and the time spent filling a pool. h = water height; t = time in hours

a) Complete the table of values for the graph.

Time, t (h)

Height, h (cm)

1

2

3

4

b) Write the linear equation.

c) What is the height of the water after 5 hours?

d) Is it reasonable to interpolate or extrapolate values on this graph? YES NO

Give a reason for your answer._________________________________________________________________

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7. Write the coordinates for the points on each graph.

Then, write the linear equation that models each graph.

Equation: Equation:

8. Sanjay wants to know how long it takes water to go from 1C to its boiling point at 100 C. He plots his data on a graph.

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a) How long did it take for the water to reach its boiling point?

b) What was the temperature of the water after 10 min?

c) At what rate did the water temperature increase? ______ C/min.

9. The graph shows the altitude of a hot-air balloon for the first 20 min after it was released.

d) Write the linear equation that models the graph. Let a = altitude and t = time.

e) How fast is the balloon rising? Write your answer in metres per minute (m/min).

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a) What was the approximate altitude of the balloon after 15 min?

b) About how long did it take the balloon to reach an altitude of 1 km?

c) How far did the balloon rise in 1 min?

10. The equation F = C + 32 models the relationship between degrees Celsius (C) and degrees Fahrenheit (F).

a) Water boils at 100C. What is the temperature in degrees Fahrenheit? Use this equation:

F = C + 32

F = ( ) + 32

F = + 32

F =

b) Complete the table for values between –50C and 130 C.

Example Calculation:

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Celsius, C (C)

Fahrenheit, F (F)

− 50 −58

− 20

+ 10

+ 40

+ 70

+ 100

+ 130

d) Graph the table of values.

e) Water freezes at 0 C. Which axis is the line intersecting or crossing?_________________________________________________________________

f) At what temperature are the values for C and F the same?

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Lesson 4 – Solving One-Step Equations (ax=b , x

a=b , a

x=b)

To find the value of a variable, we need to do the opposite of what’s happening to it (opposite operation, eg. Subtract what’s being added, divide what’s being multiplied, etc.) to both sides of the equation.

Solve the following equations:

1) 3 x=15 2) −32=−4 x 3) p4=−9

Another method to solving algebraic equations with fractions involves clearing them out. This is done by multiplying everything with a common multiple of the denominator(s) of the fraction(s) in the equation.

1)

2 x=34

2)m3

=−25

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3)

−2 1

2m=−3 1

2

4)

r0 .28

=−4 . 5

5) −1 .2 x=−9 .48

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Word Problems:1. The formula for speed iss=d

t , where s is speed, d is distance, and t is time. The length of a Canadian football field, including the end zones, is 137.2 m. If a horse gallops at 13.4 m/s, how much time would it take the horse to gallop the length of the field? Express your answer to the nearest tenth of a second.

2. Winter Warehouse has winter jackets on sale 25% off the regular price. If a jacket is on sale for $176.25, what is the regular price of the jacket?

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Lesson 5 – Solving Two-Step Equations (ax+b=c , xa+b=c)

Step 1: clear what is outside of/away from the variable first by doing the opposite of the operation to both sides of the equation.

Step 2: clear the operation that’s left by doing the opposite of it to both sides of the equation.

1. 15 x – 3=27 2. 5−2 x=19 3. 25=4 x+13

Two-Step Equations with Fractions and Decimals (Part 1)

2 x+ 110

=35

Method One: Put all terms over common denominator Method Two: Multiply both sides of equation by a common denominator

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m3

−12=−1 3

4

Method One: Put all terms over common denominator Method Two: Multiply both sides of equation by a common denominator

0.05m+4.95=18.75

Method One: Solve as per normal two step equation Method Two: Multiply both sides of equation by 10, 100, or 1000 to remove the decimal, then solve

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Practice:

Solve for the variable in each equation.

a) b)

c) d) 4c + =

e)

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Lesson 6 – Solving Two Step Equations with Fractions and Decimals (Part 2)

−2 12

m−32=−3 1

2

r2. 8

−2 .5=−3 . 7

−1 .2+ 3 .75x

=−5 . 4

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Word Problems:

1. Colin has a long-distance plan that charges 7¢/min for long distance calls within North America. There is also a monthly fee of $5.40. One month, Colin’s total long distance charges were $36.76. How many minutes of long- distance calls did Colin make that month? Write an equation and solve this problem.

2. A cylindrical storage tank that holds 380 L of water is completely full. A pump removes water at a rate of 4.8 L/min. For how many minutes must the pump work until 200 L of water remain in the tank? Write an equation and solve this problem.

3. Alice and Tessa are the top scorers on their soccer team. This season, Alice scored 1/4 of their team’s goals and Tessa scored 1/6 of their team’s goals. Alice and Tessa together scored a total of 25 goals. How many goals did their team score? Write an equation and solve this problem.

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Lesson 7 – Solving Distributive Property Equations

3(d+0 . 4 )=−3 .9

t−15

=32

On a typical February day in Whitehorse, Yukon Territory, the daily average temperature is -13.2 °C. The low temperature is -18.1 °C. What is the high temperature?

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Lesson 8 – Solving Equations with Variables on Both

Sides [ ax=b+cx ]; [ax+b=cx+d ];[ a(bx+c )=d (ex+ f ) ]

Move all terms with the variable to one side of the equation (by doing the opposite to both sides) and combine like terms before solving algebraically.

1) 3 ( x+4.2 )=10.5 2) −65

=2−x4

Practice:1) 3 x−7=8 x 2) 4 n=−5 n+45

3) 8 x+13=15 x−2 4) 2−11 m=−2 m+21

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5) 4 (5n−7 )=10 n+2 6) 0.10 d=0.25 d−7.5

7) 4.9−6.1 d=−3.2 d−3.8 8) 12

d=34 (d−3

5 )

9) 34

(d+2 )=23(d−3) 10) 3d+1

4=3+2 d

2

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Word Problem:

Alain has $35.50 and is saving $4.25/week. Eva has $24.25 and is saving $5.50/week. In how many weeks from now will they have the same amount of money?

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