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

Key Ideas

Fill in the Blanks

Fill in the Blanks

Simplify by collecting like terms

7b + 2 + 3b

Fill in the Blanks

Simplify:

12d - 4d + d

Fill in the Blanks

Simplify:

5 + 12a + 4b - 2 - 3a

Fill in the Blanks

Simplify:

13a + 8b + 2a - 5b - 4a

Fill in the Blanks

Simplify:

12uv + 7v -3vu + 3v

​Multiplying, Dividing, and mixed terms

Key Ideas

Key Ideas

Key Ideas

Fill in the Blanks

Write 4 X a X b X c without multiplication signs

Fill in the Blanks

Simplify 4a X 2b X 3c

Fill in the Blanks

Simplify 3w X 4w

​Expanding Brackets

Key Ideas

Key Ideas

Fill in the Blanks

Repeat the expression that is inside the brackets and then collect the like terms. Th number outside the brackets is the number of repeats:

2(a + k)

Fill in the Blanks

Repeat the expression that is inside the brackets and then collect the like terms. Th number outside the brackets is the number of repeats:

3(2m + 5)

Fill in the Blanks

Write two equivalent expressions for the area for the rectangle shown, only one of which contains brackets.

Fill in the Blanks

Write two equivalent expressions for the area for the rectangle shown, only one of which contains brackets.

Fill in the Blanks

Write two equivalent expressions for the area for the rectangle shown, only one of which contains brackets.

​Applying Algebra

Fill in the Blanks

The perimeter of a rectangle is given by the expression 2l + 2b, where l is the length, and b is the breadth.

Find the perimeter of a rectangle if l = 5 and b = 7.

Fill in the Blanks

The perimeter of a rectangle is given by the expression 2l + 2b, where l is the length, and b is the breadth.

Find the perimeter of a rectangle with the length 8cm and the breadth 3cm

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

Key Ideas

Fill in the Blanks

Evaluate the following expression using a = 3 and b = -5

2 + 4a

Fill in the Blanks

Evaluate the following expression using a = 3 and b = -5

7 - 4b

Fill in the Blanks

Evaluate the following expression using a = 3 and b = -5

b/5 - a

​Number patterns

Key Ideas

Fill in the Blanks

What is the rule for this number pattern:

6, 18, 30, 42, ...

Fill in the Blanks

What is the rule for this number pattern?

99, 92, 85, 78, ...

Fill in the Blanks

What is the rule for this number pattern?

2, 6, 18, 54, ...

Fill in the Blanks

What is the rule for this number pattern?

256, 128, 64, 32, ...

​Tables and Rules

Key Ideas

Fill in the Blanks

Complete the output section of the table.

Fill in the Blanks

Complete the output section of the table.

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

Key Ideas

Key Ideas

Fill in the Blanks

What are the coordinates of point A?

Remember: write coordinate pairs in inside a set of parentheses, separated by a comma - e.g. (a,b)

Fill in the Blanks

What are the coordinates of point B?

Remember: write coordinate pairs in inside a set of parentheses, separated by a comma - e.g. (a,b)

Fill in the Blanks

What are the coordinates of point C?

Remember: write coordinate pairs in inside a set of parentheses, separated by a comma - e.g. (a,b)

Fill in the Blanks

The rule for the table is

output = input + 1.

The first coordinate pair from the table is (0,1). Write the next three coordinate pairs.

​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):