Class 8: Linear Equations (Lecture Notes)

Equation: A statement of equality of two algebraic expressions involving one or more variables is called an equation.

Linear Equation: An equation of degree 1 containing one or more variables, is called a linear equation.

Simple Linear Equations (Linear Equations in One Variable): An equation of degree 1 containing only one variable is called a simple linear equation.

Some examples of simple linear equations are:

Example:

$2x+3 = 5x+2$

$2(x-y) = 4$

$\displaystyle \frac{x-1}{3} + \frac{y+2}{3} = \frac{2}{3}$

Mathematically, an equation which can be expressed in the form

$ax+b = 0, \ where \ a\ne 0$

is called a simple linear equation in the variable x. This is also called the standard form of a simple linear equation.

Solution (or Root) of an Equation: Any value of the variable which when substituted in an equation makes its left hand side (L.H.S) equal to its right hand side (R.H.S.) is called a solution (or root) of the equation.

Thus, $y$ is a root of the equation $ax+b=0 \ ay+b=0$

Therefore, solving an equation means finding all its solutions (or roots).

Note: In a simple linear equation, there is one solution for the variable. We will see that in equations of higher degree, there is more than one solution (root).

Basics rules that apply to equations:

Same number (or expression) can be added to both sides of an equation without altering the solution.
Same number (or expression) can be subtracted from both sides of an equation without altering the solution.
Both sides of an equation can be multiplied by the same non-zero number (or expression) without altering the solution.
Both sides of an equation can be divided by the same non-zero number (or expression) without altering the solution.

Transposition: Any term of an equation may be taken to the other side with the sign changed. This process is called transposition.

Example

$3x-5 = 2x+8$