Notes Booklet

Linear Equations 9

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The Coordinate Plane (Cartesian Plane)

There are quadrants formed on the coordinate plane.

an (x, y) identifies a position to the

the number in the ordered pair is called the

the number in the ordered pair is called the

rules : always , then (x, y)

always given in

Graphs of Linear Equations

y = mx + b(x, y) coordinates of a point on the line

m = rate of change (moving left – to – right)

b = y – intercept

slope the of the graph is called its

formula: slope (m) = Δ yΔ x = riserun = change∈vertical positionchange∈horizontal position

is always read from

the of the variable

Δ yΔ x

= ❑❑

y–intercept the where the graph

(0, ± n) the – axis

the of the equation

what remains when

x–intercept the where the graph

(± n, 0) the – axis

Graphing Equations on the Coordinate Plane

1. create a for the relation

use both and values

y = x + 2x y

++

0––

2. each point on a Cartesian grid

3. use a to each point

Identifying Linear Equations From a Graph

y = mx + b

1. determine the from any points on the line

m = Δ yΔ x = ❑❑ m = Δ yΔ x = ❑❑

2. identify the of the line

(0, ) (0, )

3. the values into the y = mx + b equation

Identifying Linear Equations From Word Statements

y = mx + b

1. identify the variable in the statement (look for “each” or “per” )

2. identify the variable in the statement

3. identify the in the statement (what remains if the independent variable is 0 )

B.C. Ferries charges $52.50 for the passenger vehicle plus $21.95 per passenger.

Each month, Sammy’s Cell Phone Service charges $9.95 access fee plus $0.25 per text message sent.

Susan earns $60 plus 6% commission selling sneakers at Sammy’s Shoe City.

Identifying Linear Equations From a Table of Values

y = mx + b

x y01 62 83 104 12

1. create a for the data

2. include -- this is the

3. determine the

4. determine the

5. calculate the (m = Δ yΔ x )

6. determine the - value for

7. the values into the y = mx + b equation

Identifying Linear Equations From Patterns

y = mx + b Determine the relation between the figure number (f) and the

number of squares (s) used to form the pattern:

Table of Values Graph

f s01234

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Isolating the Variable

Basic Rule

Whatever CHANGE is made to ONE side of an equation must ALSO be made to the OTHER side

process in which are made to an equation to it to the form

Using the Additive Inverse

use the inverse to create a of the rational number

n + 8 = –10 n – 5 = –4

Using the Multiplicative Inverse

use the to reduce the of the variable to

n6 = 12 8n = 56 45n = 16

Modelling Equations with Algebra Tiles

Additive Inverse

Multiplicative Inverse

Isolating the Variable in More Than One Step

Whatever CHANGE is made to ONE side of an equation must ALSO be made to the OTHER side

3n + 2 = –13 23n + 3 = –1

several might be needed to the variable in an equation

1. create a of the integer next to the variable

3n + 2 = –13 23n + 3 = –1

– 2 – 2 – 3 – 3

2. / to reduce the coefficient to 1

3n = –15 32 • 2

3n = –4 3 3

1n = –5 1n = –6

Isolating the Variable With Brackets

Whatever CHANGE is made to ONE side of an equation must ALSO be made to the OTHER side

several might be needed to the variable in an equation

4(n + 8) = 60

1. any brackets by -- make sure you multiply terms

4n + 32 = 60

2. create a to move the to the other side of the equality

4n + 32 = 60 – 32 – 32

3. / to isolate the variable

4n = 28 4 4

Identifying Solutions Through Graphing

Jade has $150 in her savings account and plans to save $100 each week from her job as a waitress.

Opal has $600 in her savings account and plans to save $50 each week from her babysitting.

After how many weeks will they have saved the same amount of money?

Challenges With Determining Solutions by Graphing

1. The (intersection) might be on the graph

2. The solution might be an value

3. The of the values might not allow a to be used

Isolating The Variable With Variables on Both Sides Whatever CHANGE is made to ONE side of an equation must

ALSO be made to the OTHER side

several might be needed to the variable in an equation

3(n – 2) = 5(n + 6)

1. any brackets by -- make sure you multiply terms

3n – 6 = 5n + 30

2. create a of the variable to gather the variable to one side of the equality

3n – 6 = 5n + 30 – 3n – 3n

3. create a of the to gather the constant to the side opposite the variable

– 6 = 2n + 30 – 30 – 30

4. / to isolate the variable

– 36 = 2n 2 2

Equations With Fractional Coefficients

Whatever CHANGE is made to ONE side of an equation must ALSO be made to the OTHER side

43 n = 6 1

3 a – 3 = – 34a + 1

2

to solve an equation involving fractional coefficients, terms by the of the fraction

34 • ( 4

3 n = 6 ) • 34

n = 184 = 9

2 = 412

Equations With Fractional Coefficients (Part Two)

if there is fraction, multiply terms by the

13 a – 3 = – 3

4a + 12

12 ( 13 a – 3 = – 3

4a + 12 )

4a – 36 = – 3a + 6

follow the remaining steps to isolate the variable

4a – 36 = – 3a + 6 + 3a + 3a

7a – 36 = + 6

7a – 36 = + 6 + 36 = + 36

7a = 42 7 7

1a = 6

Challenge Example

3n+22 – n+1

3 = n