Algebra is a branch of mathematics which helps to create mathematical  models of real world situations. Here symbols are used to represent  quantities.  The relationships between different quantities are expressed  by mathematical statements which involves symbols and operations. These relationships are called 'algebraic expressions'. Let’s  learn about these symbols and expressions in this chapter.

Constants

A constant is a symbol  which has a fixed numerical value. For example,  \(2\).  In the algebraic  expression, \(2x+3\), constants are \(2\)and \(3,\) as they have fixed values.

Variables

A variable is a quantity  which has no fixed value. We call the symbols used in algebra as 'variables', because  numbers that can be represented by these letters can 'vary'. That is, we  can assign any value for these symbols (variables). Variables are generally represented by letters by \(x,\ y,\ z,\ a,\ l\) etc. 
Consider \(3+x\).    Here the value of \(3\) cannot be changed. Hence, it is the constant. We can assign any value for the  variable \(x\). Therefore \(x\) is the variable.

Combination of a  constant and variable

Let’s consider the  expression, \(2x+5\). We know that 5 is a constant and \(x\) is the variable used in  this expression. As  \(x\) is the variable, \(x\) can be given any numerical value.    Let us assign certain  values for \(x\);            
    \begin{tabular}{cccc}    
{$Value of \(x\)$} & {$\\{\underline{\Value of \(2x\) }}(m)\}$} & {$-\Im\{\underline{\mathfrak{X}}(m)\}$} & {$\mathfrak{X}(m)$} &    {$\frac{\mathfrak{X}(m)}{23}$} & {$A_m$} & {$\varphi(m)\ /\ ^{\circ}$} & {$\varphi_m\ /\ ^{\circ}$} \\ \midrule    1  & 16.128 & +8.872 & 16.128 & 1.402 & 1.373 & -146.6 & -137.6 \\    2  & 3.442  & -2.509 & 3.442  & 0.299 & 0.343 & 133.2  & 152.4  \\    3  & 1.826  & -0.363 & 1.826  & 0.159 & 0.119 & 168.5  & -161.1 \\    4  & 0.993  & -0.429 & 0.993  & 0.086 & 0.08  & 25.6   & 90     \\    5  & 1.29   & +0.099 & 1.29   & 0.112 & 0.097 & -175.6 & -114.7 \\    6  & 0.483  & -0.183 & 0.483  & 0.042 & 0.063 & 22.3   & 122.5  \\    7  & 0.766  & -0.475 & 0.766  & 0.067 & 0.039 & 141.6  & -122   \\    8  & 0.624  & +0.365 & 0.624  & 0.054 & 0.04  & -35.7  & 90     \\    9  & 0.641  & -0.466 & 0.641  & 0.056 & 0.045 & 133.3  & -106.3 \\    10 & 0.45   & +0.421 & 0.45   & 0.039 & 0.034 & -69.4  & 110.9  \\    11 & 0.598  & -0.597 & 0.598  & 0.052 & 0.025 & 92.3   & -109.3 \\\end{tabular}
Value of x            Value of 2x            Value of 5            Value of 2x+5                  1            2            5            7                  2            4            5            9                  -1            -2            5            3                  -2            -4            5            1            
In this table we can see  the column under \(2x\) and the column under  \(2x+5\ \) vary as \(x\ \) varies, but the column  under \(5\) doesn’t vary, it remains the same. This means the combination of a  constant and variable has different values as we substitute different values for  the variable. Hence, the combination of variable and constant is also a  variable.         Illustrative examples:    Example 1:    Separate the constants  and variables from the following:         1.       5x+6    The constants are 5 and  6 and the variable used is x.         2.       7y    The constant is 7 and y  is the variable         Example 2:    If 3 is a constant and  4y is a variable, then check whether 3+4y is a constant or variable.    Solution:    We know that y can  accept different values as y is a variable. Let’s check whether 3+4y is different  for different values for y or not               Value of y            1            2            3            4                  Value of 4y            4            8            12            16                  Value of 3+4y            7            11            15            19             We can see the values of  3+4y are different for different values of y. Hence we can conclude that 3+4y  is a variable.         Example 3:     Find the values of  2-x, substituting the values of x=0, 1 and -1         Solution:    We can substitute the  value of x in 2-x;    When x=0, the value of  2-x= 2-0=2    Similarly when x=-1, we  have 2-x=\( 2-\left ( -1 \right )\)                                                            =  2+1 =3    And when x=1, 2-x= 2-1=1    Thus we have; x=0, 2-x=2    x=1, 2-x=1    x=-1, 2-x= 3              Quiz:            Fill in the blanks:      i. In 8y, ---- is the  constant and ----is the variable    ii. A quantity which is  capable of assuming many values is called -------------    iii. ------- is a  quantity whose value remains same and never can be changed.         Ans:    i. 8, y    ii. Variable    iii. Constant              2. Separate the  constants and variables from the following:    i. 2x+9    ii. 6y         Ans:                    Expression            Variables            constants                  2x+9            x            2, 9                  6y            y            6                       3. State whether the  statements are true or false (use the letter “T” for the true statement and “F”  for the false statement)    i. 10 is a constant , 3x  is a variable and 3x-10 is a variable    ii. A quantity which  takes a fixed value is called a variable    iii. 2m, 4x, 6y are some  constants    iv. Combination of a  variable and a constant is a variable         Ans:     i. T    ii. F    iii. F    iv. T