UNIT 2: ALGEBRA – CHAPTER 7: EQUATIONS, FORMULAE AND INEQUATIONS

Chapter contents

7:02  Solving equations

7:03  Equations with pronumerals on both side

7:04  Equations with grouping symbols

7:06  Solving problems using equations

7:07 Graphing inequations

7:08  Solving inequations

Learning outcomes

Students will be able techniques to solve linear equations and simple inequalities.

Explanation

• an open mathematical sentence where the “equal” sign is replaced by an inequality sign
• The signs are:

Solving equations

• Solving equations is like balancing scales
• With equations, we know that one side is equal to the other
• The solution of the equation therefore is the value of the pronumeral that ‘balances’ the equation
• Often, solving an equation requires us to change the equation into a simpler one. We can do this by adding (+), subtracting ( – ), multiplying (×) or dividing ( ÷ ) both sides of the equation by the same number
• Look at the solutions of these two equations, nothing that both sides remain balanced because the same operation is done to both sides

Let’s do some exercises

1. x + 2=4

-2    -2

x=4-2

x=2

2.

-2        -2

×4          ×4

 

Equations with pronumerals on both side

To solve these equations we have to add or subtract pronumerals as well as numerals.

Let’s do some exercises

1. 3x+1=x+5

-x     -x

2x+1=5

-1  -1

2x=4

x=2

2. 6m-1=15-2m

+2m            +2m

8m-1=15

+1 +1

8m=16

 m=2

 

Equations with grouping symbols

We need to expand the grouping symbol.

Let’s do some exercises

1. 2(x+3)=4

2x+6=4

-6 -6

2x=-2

x=-1

2. 3(a+2)=2(a+1)

3a+6=2a+2

-2a      -2a

a+6=2

-6    -6

 a=-4

 

Solving problems using equations

The main use of equations is to help us solve problems. Follow the steps bellow to do it:

Let’s do some exercises

1. The sum of a certain number and 7 is 52. What is the number?

Let’s say the number is n

n+7=52

-7   -7

n=45

2. The product of a certain number and 7 is 63. What is the number?

Let’s say the number is a

7a=63

a=9

 

Graphing inequations

Inequations usually have more than one solution.

Let’s do some exercises

1. Graph the inequation

x+3<6

x<3

2. Write an inequation to describe the graph

m<28

 

Solving inequations

Let’s do some exercises

1. 7x>14

÷7       ÷7

x>2

2. x-14 >18

+14    +14

x>32

 

So that’s all that I can tell you about Chapter 7: equations, formulae and inequations. Thank you for reading.

References: Mcseveny, A., Conway, R., Wilkes, S., & Smith, M. (2007). International Mathematics 2 for the Middle Years. In A. Mcseveny, R. Conway, S. Wilkes, & M. Smith, International Mathematics 2 for the Middle Years (pp. 152-172). Pearson Australia.

UNIT 2: ALGEBRA – CHAPTER 6: PATTERN AND ALGEBRA

Chapter contents:

6:02  Addition and Subtraction of Like Terms

6:03  Multiplication of Pronumerals

6:04  Division of Pronumerals

6:05  Multiplication and Division of Pronumerals

6:06  Using Algebra

6:08  Grouping Symbols

6:10   Algebraic Fraction

Learning outcomes

Students will be able to:

• Use letters to represent numbers and translate between words and algebraic expression.
• Create, record, analyse and generalise number patterns using words and algebraic symbols in a variety of ways.
• Use the algebraic symbol system to simplify, expand and factorise simple algebraic expressions.

Explanation

Mathematical terminologies

• Algebra (Algebraic) : the use of pronumeral when solving problem
• Expand : way to remove grouping symbol
• Expression : algebraic form that consist of numbers, variables and operation signs
• Grouping symbols : a symbol that is used to group numbers. Usually are parentheses ( ), brackets [  ] and brace {  }
• Like terms : algebraic expression that have identical pronumeral
• Unlike terms : algebraic expression that have different pronumeral part
• Pronumeral : a symbol/letter to represent unknown number
• Constant: number without variable

Addition and subtraction of like terms

When adding or subtracting terms in algebra, the sign belongs to the term after it.

Only like terms can be added or subtracted together. We can combine like terms into a single term.

Let’s do some exercises

1. 8x-2y+3x=8x+3x-2y=11x-2y 

    

2. 2a²-3bc+6a²=2a²+6a²-3bc=8a²-3bc

 

Multiplication of pronumerals

A term like ab means a X b, so when we multiply, say 3a X 6b we just need multiply the numbers together and then the pronumerals without any multiplication signs.

Let’s do some exercises

1. 2aX4=8a
2. 4pX2pX6q = 8p²6q

 

Division of pronumerals

If a division is written as one term “over” the other, we just need to divide the numbers and then cancel the pronumerals that are common in the ‘top” and “bottom”.

Let’s do some exercises

2.   

           

          

Multiplication and division of pronumerals

This is the combination of multiplication and division. The ways are still the same.

Let’s do some exercises

1. 9bX4÷2b

= 36b÷2b

= 18b

2.

           

= 12b

 

Using algebra

We can use algebra to solve word problems using the methods of addition and subtraction of like terms, multiplication of pronumerals  and division of pronumerals.

Let’s do some exercises

1. If Andy bought 2 apples which cost b cents each and 3 pears which cost 2b cents each, find an expression for the total cost

Cost of apples=2Xb                                      Cost of pears=3X2b

=2b cents                                                       =6b cents

Total cost=2b+6b=8b cents

2. If Mary bought y apples which costs 5 cents each and she had 20 cents, how much apples she can bought?

y=20÷5

y=4 apples

Grouping symbol

To write an expression without grouping symbols, we multiply each term inside the grouping symbols by the term outside.

Let’s do some exercises

1. 4(3x+2y)=4×3x+4×2y

=12x+8y

2. 2(k+5)+4k

=2k+10+4k

=6k+10

 

Algebraic fraction

It still has the same method as the regular fraction.

Let’s do some exercises

2.

 

So that’s all that I can tell you about Chapter 6: pattern and algebra. Thank you for reading.

References: Mcseveny, A., Conway, R., Wilkes, S., & Smith, M. (2007). International Mathematics 2 for the Middle Years. In A. Mcseveny, R. Conway, S. Wilkes, & M. Smith, International Mathematics 2 for the Middle Years (pp. 122-141). Pearson Australia.