Algebra techniques

​Equivalent Algebraic Expressions

By Paul Carmody

Key Ideas

Multiple Choice

Which of these expressions are equivalent:

a) 3x + 4

b) 8 - x

c) 2x + 4 + x

• a,b

• a,c

• b,c

• a,b,c

• none

​Like Terms

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​Multiplying, Dividing, and mixed terms

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​Expanding Brackets

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​Applying Algebra

Multiple Choice

Which could be the expression for the total cost, in dollars, of 10 bottles, if each bottle costs $x.

• 10 + x

• x¹⁰

• 10x

• 10/x

Multiple Choice

The total cost in dollars, of hiring a plumber for n hours. The plumber charges a $30 call-out fee plus $60 per hour

• 30 + 60n

• 90n

• 60n - 30

• 1800n

Multiple Choice

A plumber charges a $60 call-out fee plus $50 per hour. Use an expression to find out how much an 8 hour job costs.

• cost = 60 X 50n

  60 X 50 X 8 = 24,000

• cost = 60 + 50 + n

  60 + 50 + 8 = 118

• cost = 60 + 50n

  60 + 50 X 8 = 460

• cost = 50n - 60

  50 X 8 - 60 = 340

​Substitution involving negative numbers and mixed operations

ac - bc, (a = -2), (b = 5), (c = -7)

​(-2 X -7) - (5 X -7) = 14 - (-35)
= 14 + 35
= 49

ac - bc, (a = -2), (b = 5), (c = -7)

​(-2 X -7) - (5 X -7) = -14 - (-35)
= -14 + 35
= 35 - 14
= 21

ac - bc, (a = -2), (b = 5), (c = -7)

​(-2 X -7) - (5 X -7) = 14 - (-35)
= 14 - 35
= -21

ac - bc, (a = -2), (b = 5), (c = -7)

​(-2 X -7) - (5 X -7) = -14 - (-35)
= -14 - 35
= -49

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​Number patterns

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​Tables and Rules

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Multiple Choice

Which rule describes the relationship between input and output

• output = input + 9

• input = output + 9

• output = input X 4

• input/3 = output

Multiple Choice

Which rule describes the relationship between input and output

• output = input + 7

• input = output + 7

• output = input X 7

• input/7 = output

​The Cartesian Plane and Graphs

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​Let's solve some equations:

​(2p - 1) + 1 = (5) + 1

​2p - 1 + 1 = 5 + 1

​2p = 6

​(2p) ÷ 2 = (6) ÷ 2

​(2p) ÷ 2 = (6) ÷ 2

​p = 3

​b = 1

​d = 4

​Multiple variables

​y = 3 - 2x

​(2p) ÷ 2 = (6) ÷ 2

​(2p) ÷ 2 = (6) ÷ 2

​10 + (-8) = 2, t = -4

​2k = -18, k = -9

​2x = -3, x = -3/2

​x = 6

​x = 8

​x = 5

x = -48

d = -5

​b = 6

p = 11

r = 8

​x = 3
x = 0

Squares, square roots, cubes, cube roots.

​6 X 6 can be written as 6² . This is called a square.

​The square root of a number is the non-negative number that, when multiplied by itself, produces the given number. For example the square root of 36 is 6, because 6 X 6 = 36.

It is written √36 = 6

​4 X 4 X 4 can be written as 4³ . This is called a cube.

The cube root of a number is the non-negative number that, when multiplied by itself twice, produces the given number. For example the cube root of 64 is 4, because 4 X 4 X 4 = 64.

It is written ∛64 = 4

When multiplying squared numbers:
3² X 3² = 3 X 3 X 3 X 3
= 3⁴

Now make a rule for any number:
x² X x² = x⁴

What happens if I multiply the square of a number x by the cube of that same number?

​x² X x³ = (x X x) x (x X x X x)
= x⁵


What about x² X y³?
x² X y³ = x X x X y X y X y
= x²y³

​What if x has a coefficient?

2x² X x² =




7x³ X 9x⁷ =




2x² X 3y⁴ X 4x² =

​We can also divide with indices (powers/exponents):