Transcript Algebra - Eircom

Algebra
Lesson 1
Junior Certificate Mathematics
Topics To be Covered
The Use of Letters
Translating from a spoken
phrase to an Algebraic
Expression
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
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:
x  7  11
Do you know what the number is?
The answer is 4
Algebra helps us to solve
problems.
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Some examples
(i)
x  2  12 x  10

(ii)
x2  4
x6
The x here has different values in each question, we call this
letter a variable.
Notation: We write 3x instead of 3 x
Also 3 xy instead of
3 x  y
Here we simply remove the
multiplication symbol.
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
Translating from a spoken phrase to an
algebraic expression.
x plus 2 times y
x  2y
4 times x minus 8
4x  8
x decreased by 9
x 9
6 less than 3 times k
3k  6
Try Exercise From Text Book
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Algebra
Lesson 2
Junior Certificate Mathematics
Topics To be
Covered
Replacing Letters with
Numbers
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
Replacing letters with numbers
When x  3, the value of 4 x is 4  3  12
When x  2, the value of 3x  7 is 3  2  7  13
Ex 1
If x  3 and y  5, find the value of
(i) 4 x  2 y (ii) 3x  3 y (iii) 2 x  3 y  5
(i) 4 x  2 y  4  3  2  5  12  10  22
(ii) 3x  3 y  3  3  3  5  9  15  6
(iii) 2 x  3 y  5  2  3  3  5  5  6  15  5  14
Remember
Multiply and divide before you add and
subtract
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
Try the following yourself
Ex 2
If x  4 and y  2, find the value of
(i) 4 x  2 y (ii) 3x  3 y (iii) 2 x  3 y  5
(i) 4 x  2 y  4  4  2  2  16  4  20
(ii) 3x  3 y  3  4  3  2  12  6  6
(iii) 2 x  3 y  5  2  4  3  2  5  8  6  5  3
Remember
Any number when multiplied by 0 is 0,
e.g. 12 x 0 = 0
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
Try Exercise From Text Book
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Algebra
Lesson 3
Junior Certificate Mathematics
Topics To be Covered
Adding and Subtracting like
Terms
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
Adding Like Terms
2 cars + 2 cars = 4 cars
Let x be a car then this is
2x + 2x = 4x
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
Unlike Terms cannot be added
2 cars + 1 truck
Let x be a car and y be a truck
= 2x + y It cannot be reduced further
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Like Terms can be added and subtracted.
 Unlike Terms cannot be added together or
subtracted from each other.

3x  2x  5x These are like terms.
y  3 y  4 y These are like terms.
5 x  4 y These are unlike terms
and cannot be added together
5x + 4y are left just as they are
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
Simplifying Algebraic Expressions
Ex 1
Simplify 3x  4  5 x  6
3x  4  5x  6  8x 10
Try Exercise From
Text Book
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Algebra
Lesson 4
Junior Certificate Mathematics
Topics To be Covered
Removing Brackets
Removing Brackets &
then Simplifying
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
Removing Brackets
Method 1
5(2  1)  5(3)  5  3  15
Here we did what was inside the brackets first.
Method 2
52  1  52  51  10  5  15
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.
(i) 2( x  4)  2( x)  2(4)  2 x  8
(ii) 5(3 x  4)  5(3x)  5(4)  15 x  20
Everything inside the brackets is
multiplied by the number outside
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
Removing Brackets
Ex 1
4(3a  2b)  12a  8b
Ex 2
6( 2 x  3 y )  12 x  18 y
Ex 3
 3( x  2 y )  3( x)  3(2 y )
 3 x  6 y
A minus outside the bracket
changes the sign of each term
inside the bracket
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
Try the following yourself
Q1
 (3a  2b)  3a  2b
Q2
2( x  3 y )  2 x  6 y
Q3
 2(3 x  4 y ) 2(3 x)  2(4 y )
 6 x  8 y
Like signs when multiplies together give
a positive result.
Unlike signs when multiplied together give
a negative result.
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
Remove Brackets and Simplify
Ex 4
2( x  2 y  4)  3(2 x  3 y  2)
 2x  4 y  8  6x  9 y  6
 4 x  13 y  14
Try Exercise From Text Book
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Algebra
Lesson 5
Junior Certificate Mathematics
Topics To be Covered
Multiplication involving
Powers
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
Multiplication involving Powers
You know that 6  6  6 2
Do you know what 6  6  6  63
Likewise a  a  a 2
4
Do you know what a  a  a  a  a
2
When we look at a , the number 2
is called the power or index. (pluarl indices)
The power is the number at the
top of the letter
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
Multiplication involving Powers (cont.)
It is important to know that a  a1
When multiplying two similar letter together
we add the powers
2
2
4
2
3
e.g. a  a  a and a  a  a
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.
5
2
3
2
3
e.g. 2a  3a  6a
Try 7b  3b  21b
Multiply the numbers in front of
letters and add the powers at the
top of the letters.
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
Multiplication involving Powers (cont.)
Ex 1
2a (3a  2)  6a 2  4a
Ex 2
3x(2 x  3)  6 x 2  9 x
You have a new
substitute teacher today
as PowerPoint Sir is not
well.
Ex 3
2 x( x  2)  5 x( x  2 x)
2
3
2
 2 x  4 x  5 x  10 x
 5 x 3  12 x 2  4 x
2
Bring like terms together, and
reorder with highest power terms first
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
Try the following yourself
Q1
5a(2a  3)  10a  15a
2
Q2
2 x(5 x  7)  10 x 2  14 x
Q3
2 x(3x  3)  4 x( x  x)
2
3
2
 6x  6x  4x  4x
 4 x 3  10 x 2  6 x
2
Try Exercise From Text Book
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Algebra
Lesson 6
Junior Certificate Mathematics
Topics To be Covered
Calculating
Expressions with x2
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
Calculating Expressions with x2
When x  2 then x  2  2  4
2
If x  4 what does x  16
Likewise when y  3, y 2   3 3  9
2
2
y
 16
Do you know what y is when y   4
2
2
If a  3 then 4a  4  3  4  9  36
2
If a  2 then 4a 2  4   22  4  4  16
Here we have squared -2 first, then multiplied by 4.
For a x2 you always find x2 before
you multiply by a.
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
Calculating Expressions (cont.)
Ex 1 Find the value of:
3x 2  10 x  4 when x  2
 32   102   4  34   20  4
 12  20  4  4
2
Nice to be back
Ex 2 Find the value of
x  y when x  2 and y  4
2
2
 2  4  4  16  20
2
2
Here we had to square both x and y
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
Try the following yourself:
Q 1 Find the value of:
3x 2  10 x  4 when x  3
 33  103  4  39   30  4
 27  30  4  1
2
Well done
Q 2 Find the value of
2 x 2  3 y 2 when x  2 and y  - 4
 22   3 4   24   316 
 8  48  56
2
2
Remember to work out the power before you
multiply. Try Exercise From Text Book
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Algebra
Lesson 7
Junior Certificate Mathematics
Topics To be Covered
Multiplying Compound
Expressions
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
Multiplying Compound Expressions
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
Multiply each term in the second
bracket by each term in the first
bracket
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EX 1 Multiply
x  4x  5
xxxxx555( x4(x(5x) 
5)5)
x 2  5 x  4 x  20
x 2  9 x  20
When the highest power of x is 2.
This is called a Quadratic
Expression
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EX 2 Multiply
x  3x  7
xxxxx777 (x3(x(7x) 
7)7)
x 2  7 x  3x  21
x 2  10 x  21
Why is the resulting expression called a quadratic expression?
Answer: The highest power of x is 2.
When the highest power of x is 2.
The Expression is called an
Quadratic Expression?
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EX 3 Multiply
2x  3x  7
22xxxx7x7(7x3(7x()x 7) 7)
2 x 2  14 x  3x  21
2 x 2  11x  21
What is the highest power of x in the expression?
Answer: The highest power of x is 2.
When the highest power of x is 2.
The Expression is called an
Quadratic Expression?
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Q 1 Try the following yourself. Multiply:
x  3x  4
xxxxx444 (x3(x(4x) 
4)4)
x 2  4 x  3x  12
x 2  7 x  12
Try Exercise From Text Book
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