8th Class Mathematics Linear Equations in One Variable

 Linear Equations in One Variable

• Equation: An equation is a statement of equality of two algebraic expressions involving one or more unknown quantities (variables).

• Linear equation in one variable: If an equation involves only one variable and the highest index of power of that variable is 1, the equation is called a linear equation in one variable.

The general form of a linear equation in variable x is

a\[x\]+ b = 0, a\[\ne \]0 or px = q, p \[\ne \]0

• Laws of Equality

(i) The same quantity may be added to or subtracted from both sides of an equation without changing the equality.

Thus, if a = b,

a +c= b +c

a - c= b – c

(ii) If a = b then a - b = 0 (or b - a = 0).

That is, given an equality any term from one side may be transfered to the other side by changing its sign. (Law of transposition)

(iii) lf a= b then ac = be \[\frac{a}{c}=\frac{b}{c}\],(c\[\ne \]0).

That is, given an equality, both the sides can be multiplied by the same number or divided by the same nonzero number.

If \[\frac{a}{c}=\frac{b}{c}\]then multiplying both sides by bd we have ad = bc. (rule of crosswise multiplication)

(iv) If ac = be a = b provided c\[\ne \] 0. (Law of cancellation)

That is, both sides of an equality can be divided by the same nonzero number.

• The solution of a linear equation may be any rational number.

• The expressions forming equations have to be simplified before solving them. Some equations may not be linear but can be brought to a linear form by multiplying both its sides by a suitable expression.