AlgebraAlgebraLesson 1Lesson 1
Junior Certificate Mathematics
1
Topics To be Covered
The Use of Letters
Translating from a spoken phrase to an Algebraic Expression
21 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
21 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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AlgebraAlgebraLesson 2Lesson 2
Junior Certificate Mathematics
5
Topics To be Covered
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
821 April 2023
AlgebraAlgebraLesson 3Lesson 3
Junior Certificate Mathematics
9
Topics To be Covered
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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AlgebraAlgebraLesson 4Lesson 4
Junior Certificate Mathematics
14
Topics To be Covered
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.
Try the following yourself
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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Try Exercise From Text BookTry Exercise From Text Book
Remove Brackets and Simplify
18
Ex 4
)232(3)42(2 yxyx696842 yxyx
14134 yx
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AlgebraAlgebraLesson 5Lesson 5
Junior Certificate Mathematics
19
Topics To be Covered
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
20
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.
21 April 2023
Try Exercise From Text BookTry Exercise From Text Book
Try the following yourself
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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AlgebraAlgebraLesson 6Lesson 6
Junior Certificate Mathematics
24
Topics To be Covered
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
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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
21 April 2023
AlgebraAlgebraLesson 7Lesson 7
Junior Certificate Mathematics
28
Topics To be Covered
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?
21 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?
21 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
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