WEEK 2
SECTION 2.1
SRABASTI DUTTA
VARIABLES AND CONSTANT IN AN ALGEBRAIC EXPRESSION
•In math, the last few alphabets of English language are used to represent variables.
•Thus, x, y, z, usually represent variables.
•The first four letters, a, b, c, d, are used to represent constant numbers.
•If the above lettering-system is not used, then the equation will explicitly mention which
letters are used in representing the constant numbers. The other letters will then be representing
variables.
•As the name suggests, variables mean the values will be changing constantly (depending upon
the situation or circumstances); constant means a fixed value (a fixed number).
•Example: My Border-Collie Mix (a large dog) always eats twice the amount of that JackRussell Mix (a small dog). So, if I use x to represent whatever JR is eating, then BC is always
eating 2x. So, here x is the variable as it is changing all the time depending upon lunch, dinner,
treat, breakfast, etc; however 2 remains fixed.
•So, using the idea of the example, I can say that big-dogs eat ax that of small dogs, where a
can be any number based on if the pair is JR and BC or German Shepherd and Poodle (for this
pair it might be 3x) or Golden Retriever and Daschund (for this it could be 5.2x) and so on.
ALGEBRAIC EXPRESSION
An algebraic expression is any expression that consists of some
variables and constant numbers.
Some examples are: 2x + 3, 4x + 10y, x2 – 5y2 + 23.
An algebraic equation is any expression that will also include
the equal to (=) sign.
Examples are 2x + 3 = 10; 4x + 10y = -21; x2 – 5y2 + 23 = 2x +
10y
LINEAR EQUATION
•An algebraic equation of the form ax = b is known as a linear equation.
•It is called a linear equation because the graph (about which we will
learn later on) of the equation is always a line. Linear is the adjective of
line.
•a and b are the constant numbers; x is the variable.
•Examples can be 2x = 3 or -10x = 20 and so on (you can create your
equation using any kind of number for a and b).
THE ADDITION PROPERTY OF EQUALITY
Adding the same number on both sides of the equation does not change the
equation. That is, a = b implies a + c = b + c.
Example: 4 = 4. Add 3 to both sides, and you have 4 + 3 = 4 + 3  7 = 7.
Example: 2x = 4y. Add -4 to both sides to get 2x – 4 = 4y – 4, that is, the
equality holds.
You might be wondering what’s the use of this property? This property is used
in solving algebraic equation. This property tells us that if you add/subtract one
number to one side of the equation, you should also do the same on the other
side so that we are all the time comparing apples to apples and not apples to
oranges.
Correct: 2x + 3 = 10  2x + 3 – 3 = 10 – 3
Wrong: 2x + 3 = 10  2x + 3 – 3 = 10
MULTIPLICATION PROPERTY OF EQUALITY
Similarly we have a multiplication property of equality which states that
multiplying an equation, on both sides, by the same number does not change
the equality. That is a = b implies ac = bc.
Example: 2 =2  2*3 = 2*3  6 = 6
Example: 2x = 10  2x*3 = 10*3  6x = 30
This property is also used in solving equation.
1
1
2
x


10

Example: 2x = 10  2
2  x = 5 and this is the correct operation.
1
This is wrong operation: 2x = 10 2 x  2  10  x = 10. This is wrong
because here ½ was not multiplied to both sides. You can also understand
that this is wrong because if 2x = 10, then x has to be 5 so that you have
2*5 = 10.