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

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