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