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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