Manipulating Variables

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.