What are linear equations and linear inequalities?

Definition: Any algebtraic expression whose degree is one is a linear expression.

A linear equation in one variable x is of the form:

5x+9=14 or

x-3=0 or

in general y(x) = ax +b, where a is b are real.

Equations in 2 variables say x and y:

2x+3y+9=0,

x+y=6 ,

y=mx+c , slope and intercept form , m and c are any  number, and

ax+by+c=0 a standard form,

are all linear equations in two variables x, and y.

Note: x^2+y^2 = 9   or xy+x+y =k  are not  linear equations as the algebraic expression is of 2nd degree. The epression involving sqrtx or sqrty type of terms can not be linear expressions.

In general,

a1x1+a2x2+a3x3+...+an*xn =  k, where the xi (for i=1,2,...n ar n suffixes)) are n variables ans ai is ant number.

Inequalities: Inequalities are the relations  involving linear expressons. Infact equation is a special case of relation of inequality only.

example:

ax+by+c < 0  or ax+by+c>0 are inequalities, wheres

ax+by+c=0 is an equality.

ax+b<0 or ax+b>0 are inequalities whereas

ax+b=0 is an equality.

In general

a1x1+a2x2+a3x3+....+anxn < b or a1x1+a2+x2+a3x3+....+anxn>b are linear inequalities, whereas

a1x1+a2x2+a3x3+....+anxn = b is a linear equality or equation. ( Note that the highest degree of the exression is one).