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In Algebra we use alphabetical letters to describe an unknown quantity and we hope by manipulation of these letters to get the answer.

For instance when we say, what plus 3 is 8, we know it is 5. We could also have said that if x+5 = 8, what is x? In this instance x is the unknown quantity or the placeholder for the number that should have stand in the place of the x.

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x - 2 means x minus 2 , if x is 3 then x - 2 is also 3 -2 which is 1

xy means x times y , if x is 3 and y is 4 then xy is 12

x/y means x divided by y , if x is 10 and y is 2 then x/y is 5

x^2 means x times x or x to the power of two , if x is 3 then x^2 is 9

x.y means x times y

a (b+c) means a times the total given by b plus c , if a=2, b=3 and c=4

then a(b+c) = 2(3+4) = 2 times 7 = 14

2y means two times y

3wd means 3 times w times d

2 times 3y is 6y

6y divided by 5 is 6y/5

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x + 0 = x

0x = 0 times x = 0

1x is 1 times x and we write it as x, so that 1x = x as 1 times 6 = 6

x+y = y+x as in 3+4 = 4+3

xy =yx as in 3 times 4 = 4 times 3

x/x = 1 as in 4 divided by 4 is one

x/1 = x as in 6/1 is 6

0/x = 0 as in 0 divided by 10 is 0

a+(b+c) = (a+b) + c We always do the calculation in the brackets first as in

1+(2+3) = 1+5 = 6

a(b+c) = ab + ac

(a+b)(c+d) = ac + ad + bc + bd

x^n is x.x.x.x.x.x.x.x..... n times

x times -1 = -1x = -x

-x divided by -1 = x

-x/y = x/-y = -(x/y)

-x/x = -(x/x) = -1

a-b = -b + a = a + -b

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The equal sign is like the midpoint of a balancing scale. What is done on one side , must also be done on the other side to balance the scale.

The main objective in equations is to get the unknown quantity alone on one side. Usually the left side as shown in the example below.

Example

If 2x - 4 = 6 we would like the x alone on one side by doing the following

2x -4 +4 = 6+4 we plus both sides with 4 to get rid of the left side four

2x = 10 reduced form of the above

2x/2 = 10/2 we divide both sides by 2

x = 10/2 = 5 reduced form and we are left with x

and we see that 2 times 5 - 4 is indeed 6

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Example

10x -2x +6 = 4x + 8

8x + 6 = 4x + 8 reduced form of above

8x + 6 -6 = 4x + 8 - 6 add six to both sides to get rid of left side six

8x = 4x + 2 reduced form of above

8x - 4x = 4x -4x + 2 take 4x away from both sides to get right side without any form of x

4x = 2 reduced

4x/4 = 2/4 divide by 4 on both sides to get x alone

x = 1/2 = 0.5 reduced form

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We have that 2x + 3y =7 and that 4x - y = 12

We first manipulate the first equation to get x alone and then substitute that what we got into the second equation as follows

First equation

2x + 3y = 7

2x + 3y - 3y = 7 -3y minus 3y on both sides

2x = 7 - 3y reduce

2x/2 = 7/2 - 3y/2 divide by two on both sides

x = 7/2 - 3y/2 reduce

Now for the substituting

4x - y = 12 original second equation

4(7/2 - 3y/2) -y = 12 substitute the first into the second

28/2 - 12y/2 -y = 12 multiply the 4 out with the ( )

14 - 6y -y = 12 reduce

14 - 7y - 14 = 12 - 14 minus 14 on both sides

-7y = -2 reduce

-7y/-7 = -2/-7 divide by -7 on both sides

y = 2/7 reduce

Now the first equation says that 2x + 3y =7

therefore 2x + 3(2/7) = 7 by substituting y=2/7

2x + 6/7 = 7 reduce

2x + 6/7 -6/7 = 7 -6/7

2x = 6 + 1/7

2x/2 = 6/2 + 1/7/2

x = 3 + 1/14

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If we have that 3 times an unknown quantity of lemons plus 2 is 20, we substitute L for lemons and write it as follows.

3L + 2 = 20

where 3L means 3 times L.

L is therefore 6 so that 3 times 6 + 2 is 18 + 2 which is 20

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I am twice as old as my son in two years .

I am 40 years old this year.

Let me be x years now and my son be y years now.

In two years I will be x+2 and my son will be y+2.

But I will also be twice his age. Therefore 2 times (y+2) = x+2

So that 2(y+2) = x+2

And 2y + 4 = x+2

And 2y = x - 2

So two times my son's age now is also two years less than my age now.

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back to Page 2 it also contains more on Maths

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