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Math : Algebra 101

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Math : Algebra 101 [#permalink] New post 23 Sep 2010, 12:52
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Algebra 101

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This post is a part of [GMAT MATH BOOK]

created by: shrouded1


Scope

Manipulation of various algebraic expressions
Equations in 1 & more variables
Dealing with non-linear equations
Algebraic identities

Notation & Assumptions

In this document, lower case roman alphabets will be used to denote variables such as a,b,c,x,y,z,w

In general it is assumed that the GMAT will only deal with real numbers (\mathbb{R}) or subsets of \mathbb{R} such as Integers (\mathbb{Z}), rational numbers (\mathbb{Q}) etc

Concept of variables

A variable is a place holder, which can be used in mathematical expressions. They are most often used for two purposes :

(a) In Algebraic Equations : To represent unknown quantities in known relationships. For eg : "Mary's age is 10 more than twice that of Jim's", we can represent the unknown "Mary's age" by x and "Jim's age" by y and then the known relationship is x = 2y + 10
(b) In Algebraic Identities : These are generalized relationships such as \sqrt{x^2} = |x|, which says for any number, if you square it and take the root, you get the absolute value back. So the variable acts like a true placeholder, which may be replaced by any number.


Basic rules of manipulation

  1. When switching terms from one side to the other in an algebraic expression + becomes - and vice versa.
    Eg. x+y=2z \Rightarrow x=2z-y
  2. When switching terms from one side to the other in an algebraic expression * becomes / and vice versa.
    Eg. 4*x=(y+1)^2 \Rightarrow x=\frac{(y+1)^2}{4}
  3. you can add/subtract/multiply/divide both sides by the same amount. Eg. x+y = 10z \Rightarrow \frac{x+y}{43}=\frac{10z}{43}
  4. you can take to the exponent or bring from the exponent as long as the base is the same.
    Eg 1.x^2+2=z \Rightarrow 4^{x^2+2}=4^z
    Eg 2.2^{4x}=8^{y} \Rightarrow 2^{4x}=2^{3y} \Rightarrow 4x=3y

It is important to note that all the operations above are possible not just with constants but also with variables themselves. So you can "add x" or "multiply with y" on both sides while maintaining the expression. But what you need to be very careful about is when dividing both sides by a variable. When you divide both sides by a variable (or do operations like "canceling x on both sides") you implicitly assume that the variable cannot be equal to 0, as division by 0 is undefined. This is a concept shows up very often on GMAT questions.

Degree of an expression
The degree of an algebraic expression is defined as the highest power of the variables present in the expression.
Degree 1 : Linear
Degree 2 : Quadratic
Degree 3 : Cubic
Degree 4 : Bi-quadratic

Egs : x+y the degree is 1
x^3+x+2 the degree is 3
x^3+z^5 the degree of x is 3, degree of z is 5, degree of the expression is 5

Solving equations of degree 1 : LINEAR

Degree 1 equations or linear equations are equations in one or more variable such that degree of each variable is one. Let us consider some special cases of linear equations :

One variable
Such equations will always have a solution. General form is ax=b and solution is x=(b/a)

One equation in Two variables
This is not enough to determine x and y uniquely. There can be infinitely many solutions.

Two equations in Two variables
If you have a linear equation in 2 variables, you need at least 2 equations to solve for both variables. The general form is :
ax + by = c
dx + ey = f

If (a/d) = (b/e) = (c/f) then there are infinite solutions. Any point satisfying one equation will always satisfy the second

If (a/d) = (b/e) \neq (c/f) then there is no such x and y which will satisfy both equations. No solution

In all other cases, solving the equations is straight forward, multiply eq (2) by a/d and subtract from (1).

More than two equations in Two variables
Pick any 2 equations and try to solve them :
Case 1 : No solution --> Then there is no solution for bigger set
Case 2 : Unique solution --> Substitute in other equations to see if the solution works for all others
Case 3 : Infinite solutions --> Out of the 2 equations you picked, replace any one with an un-picked equation and repeat.

More than 2 variables
This is not a case that will be encountered often on the GMAT. But in general for n variables you will need at least n equations to get a unique solution. Sometimes you can assign unique values to a subset of variables using less than n equations using a small trick. For example consider the equations :
x + 2y + 5z = 20
x + 4y + 10z = 40
In this case you can treat 2y+z as a single variable to get :
x + (2y+5z) = 20
x + 2*(2y+5z) = 40
These can be solved to get x=0 and 2y+5z=20

There is a common misconception that you need n equations to solve n variables. This is not true.

Solving equations of degree 2 : QUADRATIC

The general form of a quadratic equation is ax^2+bx+c=0

The equation has no solution if b^2<4ac

The equation has exactly one solution if b^2=4ac

This equation has 2 solutions given by \frac{-b \pm \sqrt{b^2-4ac}}{2a} if b^2>4ac

The sum of roots is \frac{-b}{a}

The product of roots is \frac{c}{a}

If the roots are r_1 and r_2, the equation can be written as (x-r_1)(x-r_2)=0

A quick way to solve a quadratic, without the above formula is to factorize it :
Step 1> Divide throughout by coeff of x^2 to put it in the form x^2+dx+e=0
Step 2> Sum of roots = -d and Product = e. Search for 2 numbers which satisfy this criteria, let them be f,g
Step 3> The equation may be re-written as (x-f)(x-g)=0. And the solutions are f,g

Eg. x^2+11x+30=0
The sum is -11 and the product is 30. So numbers are -5,-6
x^2+11x+30=x^2+5x+6x+30=x(x+5)+6(x+5)=(x+5)(x+6)

Solving equations with DEGREE>2

You will never be asked to solved higher degree equations, except in some cases where using simple tricks these equations can either be factorized or be reduced to a lower degree or both. What you need to note is that an equation of degree n has at most n unique solutions.

Factorization
This is the easiest approach to solving higher degree equations. Though there is no general rule to do this, generally a knowledge of algebraic identities helps. The basic idea is that if you can write an equation in the form :

A*B*C=0

where each of A,B,C are algebraic expressions. Once this is done, the solution is obtained by equating each of A,B,C to 0 one by one.

Eg. x^3 + 11x^2 + 30x = 0
x * (x^2 + 11x + 30) = 0
x * (x+5) * (x+6) = 0

So the solution is x=0,-5,-6

Reducing to lower degree

This is useful sometimes when it is easy to see that a simple variable substitution can reduce the degree.

Eg. x^6 -3x^3 + 2 = 0
Here let y=x^3
y^2-3y+2=0
(y-2)(y-1)=0

So the solution is y=1,2 or x^3=1,2 or x=1,cube_root(3)

Other tricks

Sometimes we are given conditions such as the variables being integers which make the solutions much easier to find. When we know that the solutions are integral, often times solutions are easy to find using just brute force.

Eg. a^2+b^2=116 and we know a,b are integers such that a<b
We can solve this by testing values of a and checking if we can find b
a=1 b=root(115) Not integer
a=2 b=root(112) Not integer
a=3 b=root(107) Not integer
a=4 b=root(100)=10
a=5 b=root(91) Not integer
a=6 b=root(80) Not integer
a=7 b=root(67) Not integer
a=8 b=root(52)<a
So the answer is (4,10)

Algebraic Identities

These can be very useful in simplifying & solving a lot of questions :
  • (x+y)^2=x^2+y^2+2xy
  • (x-y)^2=x^2+y^2-2xy
  • x^2-y^2=(x+y)(x-y)
  • (x+y)^2-(x-y)^2=4xy
  • (x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)
  • x^3+y^3=(x+y)(x^2+y^2-xy)
  • x^3-y^3=(x-y)(x^2+y^2+xy)

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Re: Algebra 101 [#permalink] New post 23 Sep 2010, 12:53
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Kind of a vast topic to tackle, but I have taken a swing at it.
Suggestions for addition, editing, deletion most welcome

Please be gentle ;)
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Re: Algebra 101 [#permalink] New post 25 Sep 2010, 13:05
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wow - fantastic work and beautiful formatting!
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Re: Algebra 101 [#permalink] New post 25 Sep 2010, 16:32
Expert's post
Thanks, that's cool!
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Re: Algebra 101 [#permalink] New post 26 Sep 2010, 12:19
great stufff! kudos
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Re: Algebra 101 [#permalink] New post 27 Sep 2010, 02:37
gurpreetsingh wrote:
shrouded1 wrote:
Factorization
This is the easiest approach to solving higher degree equations. Though there is no general rule to do this, generally a knowledge of algebraic identities helps. The basic idea is that if you can write an equation in the form :

A*B*C=0

where each of A,B,C are algebraic expressions. Once this is done, the solution is obtained by equating each of A,B,C to 0 one by one.

Eg. x^3 + 11x + 30 = 0 =======> x^3 + 11x + 30x = 0
x * (x^2 + 11x + 30) = 0
x * (x+5) * (x+6) = 0

So the solution is x=0,-5,-6


thanks ! edited ...
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Re: Algebra 101 [#permalink] New post 28 Sep 2010, 20:59
Nice!!! Thanks!!!
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Re: Math : Algebra 101 [#permalink] New post 30 Sep 2010, 04:23
Good one Shrouded..Thanks !
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Re: Math : Algebra 101 [#permalink] New post 08 Oct 2010, 02:04
Nice and thanks!!

:)
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Re: Math : Algebra 101 [#permalink] New post 09 Oct 2010, 07:11
Kudos! Thanks for making a valuable contribution :).
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Re: Math : Algebra 101 [#permalink] New post 14 Dec 2010, 08:37
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Friend, I appreciate your effort!
I do believe you made a mistake in this equation, shouldn't there be 5z instead of z?

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Re: Math : Algebra 101 [#permalink] New post 15 Dec 2010, 08:09
More than 2 variables
This is not a case that will be encountered often on the GMAT. But in general for n variables you will need at least n equations to get a unique solution. Sometimes you can assign unique values to a subset of variables using less than n equations using a small trick. For example consider the equations :
x + 2y + 5z = 20
x + 4y + 10z = 40

In this case you can treat 2y+z as a single variable to get :
x + (2y+z) = 20
x + 2*(2y+z) = 40

These can be solved to get x=0 and 2y+z=20

There must be some mistake here because 2y+5z is what equals 20 not 2y+z. If it is correct please tell me how that can happen?

Thanks
Mari
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Re: Math : Algebra 101 [#permalink] New post 29 Mar 2011, 13:01
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Re: Math : Algebra 101 [#permalink] New post 05 Apr 2011, 10:55
thts really helpful for revising..
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Re: Math : Algebra 101 [#permalink] New post 05 Apr 2011, 10:57
is there an easier way to find square root of unusual number ... like 191 etc or to see if it's square root exists or not .. .. any tips and tricks?
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Re: Math : Algebra 101 [#permalink] New post 05 Apr 2011, 11:53
simmy818 wrote:
is there an easier way to find square root of unusual number ... like 191 etc or to see if it's square root exists or not .. .. any tips and tricks?


http://www.indianmathonline.com/square.pdf
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Re: Math : Algebra 101 [#permalink] New post 06 Apr 2011, 23:08
Great stuff.
Thank you!
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Re: Math : Algebra 101 [#permalink] New post 03 Jul 2011, 23:09
shrouded1 wrote:
[*](x+y)^2=x^2+y^2+2xy
[*](x-y)^2=x^2+y^2-2xy
[*]x^2-y^2=(x+y)(x-y)
[*](x+y)^2-(x-y)^2=4xy
[*](x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)
[*]x^3+y^3=(x+y)(x^2+y^2-xy)
[*]x^3-y^3=(x-y)(x^2+y^2+xy)[/list]


Can somebody give me the link for theory of Inequalities??????? :? :?
i found the modulus theory in the book but on inequity i couldn't :? :?
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Re: Math : Algebra 101 [#permalink] New post 17 Jan 2013, 08:59
Need Help!!! Urgent

What is the difference between the solution of below problems, Is there any difference in the approach:-
1. X^3 > X

soln: -1<X<0 or X>1

If we follow the above principle for solving than the answer to below equation must be:

2. a^3< a^2 + 2a

-1<a<0 or a>2

but the answer as per MGMAt quant book is
0<a<2 or a<-1

Pls help me in getting the right concept.........
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Re: Math : Algebra 101 [#permalink] New post 28 Jan 2013, 19:17
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Archit143 wrote:
Need Help!!! Urgent

What is the difference between the solution of below problems, Is there any difference in the approach:-
1. X^3 > X

soln: -1<X<0 or X>1

If we follow the above principle for solving than the answer to below equation must be:

2. a^3< a^2 + 2a

-1<a<0 or a>2

but the answer as per MGMAt quant book is
0<a<2 or a<-1

Pls help me in getting the right concept.........


Responding to a pm:

2. a^3< a^2 + 2a

a^3 - a^2 - 2a < 0
a(a - 2)(a + 1) < 0

......-1.....0............2.......

Since you want the expression to be negative, the required range is a < -1 or 0 < a < 2 (Notice the sign in the original inequality. It is less than)

The range you have written: -1<a<0 or a>2 is for the expression to be positive.

Check out these posts for further clarification:

http://www.veritasprep.com/blog/2012/06 ... e-factors/
http://www.veritasprep.com/blog/2012/07 ... ns-part-i/
http://www.veritasprep.com/blog/2012/07 ... s-part-ii/
http://www.veritasprep.com/blog/2012/07 ... qualities/
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Re: Math : Algebra 101   [#permalink] 28 Jan 2013, 19:17
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