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Maths Module 5

Algebra Basics

This module covers concepts such as:

• algebra rules
• collecting like terms
• simplifying equations

www.jcu.edu.au/students/learning-centre

Module 5

Algebra Basics

1. What is Algebra
2. Glossary
3. Some Algebra Rules
4. Addition & Multiplication Properties
5. Collecting Like Terms
6. Simplifying Equations: Using Expansion
7. Simplifying Equations with Exponents
8. Answers
9. Helpful Websites

2. Glossary

2 𝑥𝑥, 3 & 17 are TERMS 2 𝑥𝑥 + 3 = 17 is an EQUATION 17 is the SUM of 2 𝑥𝑥 + 3

3 is a CONSTANT

2 is a COEFFICIENT

𝑥𝑥 is a VARIABLE

2 𝑥𝑥 + 3 is an EXPRESSION + is an OPERATOR

Equation: Is a mathematical sentence. It contains an equal sign meaning that both sides are equivalent.

Expression: An algebraic expression involves numbers, operation signs, brackets/parenthesis and pronumerals that substitute numbers.

Operator: The operation (+ , − ,× ,÷) which separates the terms.

Term: Parts of an expression separated by operators which could be a number, variable or product of numbers and variables. Thus 2 𝑥𝑥, 3 & 17

Pronumeral: A symbol that stands for a particular value.

Variable: A letter which represents an unknown number. Most common is 𝑥𝑥, but it can be any symbol.

Constant: Terms that contain only numbers that always have the same value.

Coefficient: Is a number that is partnered with a variable. Between the coefficient and the variable is a multiplication. Coefficients of 1 are not shown.

In summary:

Pronumeral: 𝑥𝑥 Operator: + Variable: 𝑥𝑥 Terms: 3 , 2 𝑥𝑥 (a term with 2 factors) & 17 Constant: 3 Equation: 2x + 3 = 17 Left hand expression: 2 𝑥𝑥 + 3 Coefficient: 2 Right hand expression 17 (which is the sum of the LHE)

2. Your Turn:

Complete the following for the equation: 5a + 3 = 38

Pronumeral: Operator: Variable: Terms: Constant: Expression: Equation: Left hand expression: Coefficient: Right hand expression

3. Some Algebra Rules

Expressions with zeros and ones: Zeros and ones can be eliminated. For example: When we add zero it does not change the number, 𝑥𝑥 + 0 = 𝑥𝑥 or 𝑥𝑥 − 0 = 𝑥𝑥 (6 + 0 = 6, 6 − 0 = 6) If we multiply by one, then the number stays the same, 𝑥𝑥 × 1 = 𝑥𝑥 or 𝑥𝑥 1 = 𝑥𝑥 (6 × 1 = 6, 61 = 6) Note: when we work with indices (powers) any number raised to the power zero is 1, this works because when we divide indices we subtract the indexes and thus get zero. 22 ÷ 2^2 = 4 ÷ 4 = 1 𝑐𝑐ℎ𝑒𝑒𝑝𝑝𝑒𝑒𝑜𝑜𝑐𝑐𝑝𝑝𝑒𝑒, 2^2 ÷ 2^2 = 22−2^ = 2^0 = 1

• Multiplicative Property: 1 × 𝑥𝑥 = 𝑥𝑥
• Multiplying any number by one makes no difference.
• Additive Inverse: 𝑥𝑥 + (−𝑥𝑥) = 0
• Any number added to its negative equals zero.
• Multiplicative Inverse: 𝑥𝑥×𝑥𝑥^1 = 1
• Any number multiplied by its reciprocal equals one. 𝑥𝑥 × (^1) 𝑥𝑥 = 1; 4 × 14 = 1
• Symmetric Property: 𝐼𝐼𝑜𝑜 𝑥𝑥 = 𝑦𝑦 𝑐𝑐ℎ𝑒𝑒𝑛𝑛 𝑦𝑦 = 𝑥𝑥 Perfect harmony.
• Transitive Property: 𝐼𝐼𝑜𝑜 𝑥𝑥 = 𝑦𝑦 𝑝𝑝𝑛𝑛𝑎𝑎 𝑦𝑦 = 𝑧𝑧, 𝑐𝑐ℎ𝑒𝑒𝑛𝑛 𝑥𝑥 = 𝑧𝑧 For example, if apples cost $2 and oranges cost $2 then apples and oranges are the same price.

The Calculation Priority Sequence recognised by the mnemonic known as the BIDMAS of BODMAS.

Also a golden rule : “What we do to one side we do to the other”

3. Your Turn:

Here are some revision examples for practise:

a. 10 − 2 × 5 + 1 =

b. 10 × 5 ÷ 2 − 3 =

c. 12 × 2 − 2 × 7 =

d. 48 ÷ 6 × 2 − 4 =

e. 𝑤𝑤ℎ𝑝𝑝𝑐𝑐 𝑖𝑖𝑐𝑐 𝑐𝑐ℎ𝑒𝑒 𝑚𝑚𝑖𝑖𝑐𝑐𝑐𝑐𝑖𝑖𝑛𝑛𝑚𝑚 𝑐𝑐𝑝𝑝𝑒𝑒𝑝𝑝𝑝𝑝𝑐𝑐𝑖𝑖𝑐𝑐𝑛𝑛 𝑐𝑐𝑦𝑦𝑚𝑚𝑜𝑜𝑐𝑐𝑜𝑜 18 _ 3 × 2 + 2 = 14

5. Collecting Like Terms

Algebraic thinking involves simplifying problems to make them easier to solve. Often the sight of variables can raise mathematical anxieties, yet once we understand a little more, the anxieties can dissipate.

Mathematical anxiety is common and the best way to relieve it is to recognise what we do know; then begin to work methodically to solve the problem. For instance, in the equation below, we can look at the equation as bits of information so as the equation becomes easier to solve. In short, we simplify the problem into a smaller problem and this is done by collecting like terms.

7 𝑥𝑥 + 2𝑥𝑥 + 3𝑥𝑥 − 6 𝑥𝑥 + 2 = 14

A like term is a term which has the same variable (which may also have the same power/exponent/index), only the coefficient is different. For example, in the equation above we can see that we have four different coefficients (7, 2, 3, & 6) with the same variable, 𝑥𝑥, yet there are no exponents to consider. Once identified, we can collect these like terms: 7 𝑥𝑥 + 2𝑥𝑥 + 3𝑥𝑥 − 6 𝑥𝑥 We can add and subtract the coefficients as separate from the variable: 7 + 2 + 3 − 6 = 6 Thus, 7 𝑥𝑥 + 2𝑥𝑥 + 3𝑥𝑥 − 6 𝑥𝑥 = 6𝑥𝑥 The original equation simplifies to 6 𝑥𝑥 + 2 = 14 (adding in the constant) Now we solve the equation: 6 𝑥𝑥 + 2(−2) = 14 − 2 6 𝑥𝑥 = 12 6 𝑥𝑥(÷ 6) = 12 ÷ 6 𝑥𝑥 = 2 This mathematical process will be investigated more deeply in section 6.

E XAMPLE P ROBLEM :

1. Collect the like terms and simplify: 5 𝑥𝑥 + 3𝑥𝑥𝑦𝑦 + 2𝑦𝑦 − 2 𝑦𝑦𝑥𝑥 + 3𝑦𝑦 2 Step 1: Recognise the like terms (note: 𝑥𝑥𝑦𝑦 is the same as 𝑦𝑦𝑥𝑥; commutative property) 5 𝑥𝑥 + 3𝑥𝑥𝑦𝑦 + 2𝑦𝑦 − 2 𝑦𝑦𝑥𝑥 + 3𝑦𝑦 2 Step 2: Arrange the expression so that the like terms are together (remember to take the operator with the term). 5 𝑥𝑥 + 2𝑦𝑦 + 3𝑥𝑥𝑦𝑦 − 2 𝑦𝑦𝑥𝑥 + 3𝑦𝑦 2 Step 3: Complete the operation: 5 𝑥𝑥 + 2𝑦𝑦 + 𝑥𝑥𝑦𝑦 + 3𝑦𝑦 2 Note: a coefficient of 1 is not usually shown ∴ 5 𝑥𝑥 + 2𝑦𝑦 + 𝑥𝑥𝑦𝑦 + 3𝑦𝑦 2 5. Your Turn:

Collect the like terms using the steps above:

a. 3 𝑥𝑥 + 2𝑦𝑦 − 𝑥𝑥

b. 2 𝑥𝑥 2 − 3 𝑥𝑥 3 − 𝑥𝑥 2 + 2𝑥𝑥

c. 3 𝑚𝑚 + 2𝑛𝑛 + 3𝑛𝑛 − 𝑚𝑚 − 7

d. 4(𝑥𝑥 + 7) + 3(2𝑥𝑥 − 2)

e. 3(𝑚𝑚 + 2𝑛𝑛) + 4(2𝑚𝑚 + 𝑛𝑛)

f. 𝑥𝑥 3 + 𝑥𝑥 4

6. Simplifying Equations: Using Expansion

To simplify equations involves ‘expanding’ or ‘factorising’. This section helps you to investigate the concept of what it means to expand an expression. When we expand an expression, we remove the brackets, often referred to as the grouping symbols. This ‘expansion’ involves applying the distributive property.

Let’s illustrate with an example: 𝑥𝑥(6 + 9)^ As we know from the distributive law, the ′𝑥𝑥 ′^ outside of the brackets is multiplied through the brackets. So we can express 𝑥𝑥(6 + 9) as: 6 𝑥𝑥 + 9𝑥𝑥. In this expression we have two like terms, so we can simplify further to 15 𝑥𝑥.

If we multiply two numbers together, then the order in which we multiply is irrelevant; commutative property. For example: Simplify 4(3𝑥𝑥) This could be written as 4 × (3 × 𝑥𝑥) and then as (4 × 3) × 𝑥𝑥 Therefore, we can simplify to 12𝑥𝑥

6. Your Turn:

Simplify these expressions using expansion:

a. 𝑥𝑥(4 + 3)^ b. 𝑥𝑥(3 − 1)^ c. 𝑥𝑥(8 + 6)

Sometimes there may be nested grouping symbols. This happens when there are two sets of brackets – one is nested inside the other. This means that the operations in the inner set must be worked first.

E XAMPLE P ROBLEMS :

1) 20 − [3(14 − 10)] =

20 − [3 × (14 − 10)] =

20 − [3 × 4] =

2) 4[(6 + 3) × 5] =

4[9 × 5] =

3 ) [12−(3+21(11−7))] = ((3+2112−4)) ) =

24 8 = 3

6. Your Turn

d. 8[9 − (5 + 2)] =

e. 2[4 + 5(6 − 5)] =

f. 2[3(13 − 8) × 4] =

g. 2 ( 2+34+6 )=

h. ( 412 (4+2×^4 )) =

i. What is the missing number? 5 + {4[_ + 3(7 + 2)]} = 125

7. Your Turn

a. 2 𝑥𝑥 + 7𝑥𝑥 + 11𝑥𝑥 =

b. 4 𝑥𝑥𝑦𝑦 + 7𝑥𝑥𝑦𝑦 =

c. 6 𝑥𝑥 2 − 5 𝑥𝑥 2 =

d. 5 𝑥𝑥 2 + 7𝑥𝑥 + 3𝑥𝑥 =

e. 8 𝑥𝑥 2 y + 2𝑥𝑥 2 𝑦𝑦 + 5𝑥𝑥𝑦𝑦 =

f. …and a challenge: What is the missing number 2 (^ +4)

2 5 (14−3 2 ) = 8

8. Answers

1. a. 27 b. 6 c. 4

Pronumeral: 𝑝𝑝 Operator: + Variable: 𝑝𝑝 Term: 5 𝑝𝑝 & 3 Constant: 3 Expression: 5 𝑝𝑝 + 3 Equation: 5 𝑝𝑝 + 3 = 38 Left hand expression: 5 𝑝𝑝 + 3 Coefficient: 5 Right hand expression 38

3. a. 1 b. 22 c. 10 d. 12 e. –

a. 3 × 2𝑥𝑥 𝑐𝑐𝑝𝑝 2 × 3𝑥𝑥; (6𝑥𝑥 simplified) b. 2𝑥𝑥 × 8 (16𝑥𝑥 simplified) c. (8 × 2) + (8𝑥𝑥) 𝑐𝑐𝑝𝑝(16 + 8𝑥𝑥)

5. a. 3 𝑥𝑥 + 2𝑦𝑦 − 𝑥𝑥 = 2𝑥𝑥 + 2𝑦𝑦 b. 2 𝑥𝑥 2 − 3 𝑥𝑥 3 − 𝑥𝑥 2 + 2𝑥𝑥 = 𝑥𝑥 2 − 3 𝑥𝑥 3 + 2𝑥𝑥 c. 3 𝑚𝑚 + 2𝑛𝑛 + 3𝑛𝑛 − 𝑚𝑚 − 7 = 2𝑚𝑚 + 5𝑛𝑛 − 7 d. 4(𝑥𝑥 + 7) + 3(2𝑥𝑥 − 2) = 4𝑥𝑥 + 28 + 6𝑥𝑥 − 6 = 10𝑥𝑥 + 22 e. 3(𝑚𝑚 + 2𝑛𝑛) + 4(2𝑚𝑚 + 𝑛𝑛) = 3𝑚𝑚 + 6𝑛𝑛 + 8𝑚𝑚 + 4𝑛𝑛 = 11𝑚𝑚 + 10𝑛𝑛 f. 𝑥𝑥 3 + 𝑥𝑥 4 = (4𝑥𝑥+3𝑥𝑥 12 )= 7𝑥𝑥 12
6. a. 4 𝑥𝑥 + 3𝑥𝑥, 𝑐𝑐𝑝𝑝 7 𝑥𝑥 b. 3 𝑥𝑥 − 𝑥𝑥, 𝑐𝑐𝑝𝑝 2 𝑥𝑥 c. 8 𝑥𝑥 + 6𝑥𝑥, 𝑐𝑐𝑝𝑝 14 𝑥𝑥 d. 16 e. 18 f. 120 g. 4 h. 2 i. 3
7. a. 20𝑥𝑥 b. 1 1 𝑥𝑥𝑦𝑦 c. 𝑥𝑥 2 d. 5 𝑥𝑥 2 + 10 𝑥𝑥 e. 10 𝑥𝑥 2 𝑦𝑦 + 5𝑥𝑥𝑦𝑦 f. 6 9. Helpful Websites

BIDMAS: http://www.educationquizzes.com/gcse/maths/bidmas-f/ Commutative property: http://www.mathematicsdictionary.com/english/vmd/full/c/vepropertyofmultiplication.ht m http://www.purplemath.com/modules/numbprop2.htm

Like Terms: http://www.freemathhelp.com/combining-like-terms.html