Notes Booklet

Linear Equations 8

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The Coordinate Plane (Cartesian Plane)

The number line is called the axis

The number line is called the axis.

The two lines meet at a point called the .

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

Graphing on the Coordinate Plane

1. create a for the relation

use both and values

x + 2 = y Co–ordinate pairx y

0

2. each point on a Cartesian grid

3. use a to each point

Reading Graphs of Linear Equations

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

m = rate of change (moving left 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

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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Equations

equation a statement that two mathematical expressions are and have the

terms the “ ” of the math sentence

are separated by the operation signs

variables the in the expression

they can represent many values

coefficient the in front of the variable that the variable

we write the coefficient first and the variable second

Modelling Equations With Algebra Tiles

Representing Equations With Algebra Tiles

Many first-degree equations can be represented with algebra tiles

Isolating the Variable

Basic Rule

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

Using the Additive Inverse

n + 4 = 11 n – 5 = 7 n – 8 = –1 n + 3 = –13

use the inverse to create a of the integer in order to the variable

Using the Multiplicative Inverse

8n = 56 n6 = 12 3

4n = 9Basic Rule

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

use the inverse to create a coefficient of in front of the variable

5n = –15 n6 = – 3

Isolating the Variable in More Than One Step 1

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

3n + 2 = –13 several might be needed to

the variable in an equation

1. create a of the integer next to the variable

2. / to isolate the variable

Isolating the Variable in More Than One Step 2

n2 – 2 = 5

follow the same steps when dealing with a fractional variable

1. create a of the integer next to the variable

2. / to isolate the variable

Isolating The Variable With Brackets

6(t – 1) = 30

(factored form)

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

2. create a of the integer next to the variable

3. / to isolate the variable

Isolating The Variable With Brackets

Use your algebra tiles to model the steps to isolate the variable:

Show the algebraic steps next to your model:

Express the model in both:

(a) factored Form:

(b) distributed Form: