Algebra Lesson 1 Junior Certificate Mathematics 1 Topics To be Covered The Use of Letters Translating from a spoken phrase to an Algebraic Expression 02

AlgebraAlgebraLesson 1Lesson 1

Junior Certificate Mathematics

1

Topics To be Covered

The Use of Letters

Translating from a spoken phrase to an Algebraic Expression

18 April 2023

Algebra helps us to solve Algebra helps us to solve problems.problems.

In algebra we use letters to stand for numbers.I think of a number, add seven to it and the result is 11.

If we let x stand for the number we can write this as follows:

117 xDo you know what the number is?

The answer is 4

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Here we simply remove the Here we simply remove the multiplication symbol.multiplication symbol.

Some examples

3

122 x 10x

42 x 6x

(i)

(ii)

The x here has different values in each question, we call this letter a variable.

Notation: We write instead of

Also instead of xy3

x3 x3

yx3

18 April 2023

Try Exercise From Text BookTry Exercise From Text Book

Translating from a spoken phrase to an algebraic expression.

4

x plus 2 times y

4 times x minus 8

x decreased by 9

6 less than 3 times k

yx 2

84 x

9x

63 k

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5


Replacing Letters with Numbers

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RememberRememberMultiply and divide before you add and Multiply and divide before you add and subtractsubtract

Replacing letters with numbers

6

1234 is 4 of value the,3When xx

13723 is 73 of value the,2When xx

Ex 1

532 (iii) 33 (ii) 24(i)

of value thefind ,5 and 3 If

yxyxyx

yx

221012523424 (i) yx6159533333 (ii) yx

14515655332532 (iii) yx

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RememberRememberAny number when multiplied by 0 is 0,Any number when multiplied by 0 is 0,e.g. 12 x 0 = 0e.g. 12 x 0 = 0

Try the following yourself

7

Ex 2

532 (iii) 33 (ii) 24(i)

of value thefind ,2 and 4 If

yxyxyx

yx

20416224424 (i) yx

6612234333 (ii) yx

356852342532 (iii) yx

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Try Exercise From Text Book

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9


Adding and Subtracting like Terms

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2 cars + 2 cars = 4 cars2 cars + 2 cars = 4 carsLet Let xx be a car then this is be a car then this is 22xx + 2 + 2xx = 4 = 4xx

Adding Like Terms

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2 cars + 1 truck 2 cars + 1 truck Let Let xx be a car and be a car and yy be a truck be a truck = = 22xx + + y y It cannot be reduced furtherIt cannot be reduced further

Unlike Terms cannot be added

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55xx + 4 + 4yy are left just as they are are left just as they are

Like Terms can be added and subtracted.Unlike Terms cannot be added together or

subtracted from each other.

12

terms.like are These 523 xxx terms.like are These 43 yyy

togetheradded becannot and

termsunlike are These 45 yx

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Try Exercise From Try Exercise From Text BookText Book

Simplifying Algebraic Expressions

13

6543Simplify xx

Ex 1

1086543 xxx

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14


Removing Brackets

Removing Brackets & then Simplifying

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Everything inside the brackets is Everything inside the brackets is multiplied by the number outsidemultiplied by the number outside

Removing Brackets

15

1535)3(5)12(5 Here we did what was inside the brackets first. 155101525125 Here we multiplied everything inside the bracket by 5.The final result is the same as Method 1. In this section we may not be able to reduce what is inside the bracket further, so Method 2 is what we use.

)4(2 (i) x 82)4(2)(2 xx

Method 1

Method 2

)43(5 (ii) x 2015)4(5)3(5 xx

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A minus outside the bracketA minus outside the bracketchanges the sign of each term changes the sign of each term inside the bracketinside the bracket

Removing Brackets

16

Ex 1

)23(4 ba ba 812 Ex 2

)32(6 yx yx 1812 Ex 3

)2(3 yx yx

yx

63

)2(3)(3

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Like signs when multiplies together giveLike signs when multiplies together givea positive result. a positive result. Unlike signs when multiplied together giveUnlike signs when multiplied together givea negative result.a negative result.


17

Q 1

)23( ba ba 23 Q 2

)3(2 yx yx 62 Q 3

)43(2 yx yx

yx

86

)4(2)3(2

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Remove Brackets and Simplify

18

Ex 4

)232(3)42(2 yxyx696842 yxyx

14134 yx

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19


Multiplication involving Powers

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The power is the number at the The power is the number at the top of the lettertop of the letter

Multiplication involving Powers

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2666 that knowYou 666 what knowyou Do 36

2 Likewise aaa aaaa what knowyou Do 4a

indices) (pluarl index.or power thecalled is

2number the,at look When we 2a

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Multiply the numbers in front of Multiply the numbers in front of letters and add the powers at the letters and add the powers at the top of the letters.top of the letters.

Multiplication involving Powers (cont.)

21

32 e.g. aaa

1 that know toimportant isIt aa When multiplying two similar letter together we add the powers

422 and aaa When multiplying two similar letter together that have numbers in front of them we multiply these numbers infront together, and add the powers as before.

32 632 e.g. aaa 32 37Try bb 521b

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Bring like terms together, and Bring like terms together, and reorder with highest power terms firstreorder with highest power terms first

Multiplication involving Powers (cont.)

22

Ex 1

)23(2 aa aa 46 2 Ex 2

)32(3 xx xx 96 2 Ex 3

)2(5)2(2 2 xxxxx 232 10542 xxxx

xxx 4125 23

You have a new substitute teacher today as PowerPoint Sir is not well.

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23

Q 1

)32(5 aa aa 1510 2 Q 2

)75(2 xx xx 1410 2 Q 3

)(4)33(2 2 xxxxx 232 4466 xxxx

xxx 6104 23

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24


Calculating Expressions with x2

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For For a xa x22 you always find you always find xx22 before before you multiply by you multiply by aa..

Calculating Expressions with x2

25

422 then 2When 2 xx 2 does what 4 If xx 16

933 3, when Likewise 2 yy4 when is what knowyou Do 2 yy 162 y

3694344 then 3 If 22 aa

24 then 2 If aa 164424 2

Here we have squared -2 first, then multiplied by 4.

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Here we had to square both Here we had to square both xx and and yy

Calculating Expressions (cont.)

26

Ex 1 Find the value of:2 when 4103 2 xxx

442012

42043421023 2

Ex 2 Find the value of

4 and 2 when 22 yxyx

2016442 22

Nice to be back

18 April 2023

Remember to work out the power before you Remember to work out the power before you multiply. Try Exercise From Text Bookmultiply. Try Exercise From Text Book

Try the following yourself:

27

Q 1 Find the value of:3 when 4103 2 xxx

143027

43093431033 2

Q 2 Find the value of

4- and 2 when 32 22 yxyx

56488

163424322 22

Well done

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28


Multiplying Compound Expressions

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Multiply each term in the second Multiply each term in the second bracket by each term in the first bracket by each term in the first bracketbracket

Multiplying Compound Expressions

29

When multiplying compound algebraic expressions together, we multiply each term in the second bracket, by each term in the first bracket.

(i) Write down the second bracket twice.(ii) Multiply the 1st term in the 1st bracket by the 2nd bracket.(iii) Multiply the 2nd term in the 1st bracket by the 2nd bracket.(iv) Remove the brackets(by multiplying)(v) Bring like terms together

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When the highest power of When the highest power of xx is 2. is 2. This is called a Quadratic This is called a Quadratic ExpressionExpression

30

54 xx

EX 1 Multiply

)5(5 xx )5(5 xxx )5(45 xxx

20452 xxx

2092 xx

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When the highest power of When the highest power of xx is 2. is 2. The Expression is called an The Expression is called an Quadratic Expression?Quadratic Expression?

31

73 xx

EX 2 Multiply

)7(7 xx )7(7 xxx )7(37 xxx

21372 xxx21102 xx

Answer: The highest power of x is 2.

Why is the resulting expression called a quadratic expression?

18 April 2023

When the highest power of When the highest power of xx is 2. is 2. The Expression is called an The Expression is called an Quadratic Expression?Quadratic Expression?

32

732 xx

EX 3 Multiply

)7(7 xx )7(72 xxx )7(372 xxx

213142 2 xxx21112 2 xx

Answer: The highest power of x is 2.

What is the highest power of x in the expression?

18 April 2023

Try Exercise From Text BookTry Exercise From Text Book33

43 xx

Q 1 Try the following yourself. Multiply:

)4(4 xx )4(4 xxx )4(34 xxx

12342 xxx

1272 xx

18 April 2023

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Algebra Lesson 1 Junior Certificate Mathematics 1 Topics To be Covered The Use of Letters Translating from a spoken phrase to an Algebraic Expression 02