Manipulating Variables
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Variable
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined.[1] The concepts of constants and variables are fundamental to many areas of mathematics and its applications. A "constant" in this context should not be confused with a mathematical constant which is a specific number independent of the scope of the given problem.
One may use any letteras m,l,p,x,y,z etc to show a variable . Remeber , a
variable is a number which does not have a fixed value. For ex ,the
number 5 or the 100 or any other given number is not a variable. They
are fixed values.(constant). Similiarly the number of angles of a
triangle has a fixed value i e 3. It is not a variable.The number of
corners of a qudrilateral is fixed (4 ) it is also not a variable.
But the measurement of each side of a qudrilateral is not fixed.
Dependent and independent variables
Variables are further distinguished as being either a dependent variable or an independent variable. Independent variables are regarded as inputs to a system and may take on different values freely. Dependent variables are those values that change as a consequence to changes in other values in the system.
Expressions
An expression is a mathematical term or a sum or difference of mathematical terms that may use numbers, variables, or both.
Example:
The following are examples of expressions:
- 2
- x
- 3 + 7
- 2 × y + 5
- 2 + 6 × (4 - 2)
- z + 3 × (8 - z)
Example:
Gangaiah weighs 70 kilograms, and Somanna weighs k kilograms. Write an expression for their combined weight. The combined weight in kilograms of these two people is the sum of their weights, which is 70 + k.
Example:
A car travels down the highway at 55 kilometers per hour. Write an expression for the distance the car will have traveled after h hours. Distance equals rate times time, so the distance traveled is equal to 55 × h..
Example:
There are 2000 liters of water in a swimming pool. Water is filling the pool at the rate of 100 liters per minute. Write an expression for the amount of water, in liters, in the swimming pool after m minutes. The amount of water added to the pool after m minutes will be 100 liters per minute times m, or 100 × m. Since we started with 2000 liters of water in the pool, we add this to the
amount of water added to the pool to get the expression 100 × m + 2000.
To evaluate an expression at some number means we
replace a variable in an expression with the number, and simplify the
expression.
Example:
Evaluate the expression 4 × z + 12 when z = 15.
We replace each occurrence of z with the number
15, and simplify using the usual rules: parentheses first, then
exponents, multiplication and division, then addition and
subtraction.
4 × z + 12 becomes
4 × 15 + 12 =
60 + 12 =
72
Example:
Evaluate the expression (1 + z) × 2 + 12 ÷ 3 - z
when z = 4.
We replace each occurrence of z with the number 4,
and simplify using the usual rules: parentheses first, then
exponents, multiplication and division, then addition and
subtraction.
(1 + z) × 2 + 12 ÷ 3 - z becomes
(1 + 4) × 2 + 12 ÷ 3 - 4 =
5 × 2 + 12 ÷ 3 - 4 =
10 + 4 - 4 =
10.