Linear Equations

Equation of the form `ax=b`, where a and b are real numbers, is called linear equation with one variable x; a is called coefficient near variable, b is called free member.

For linear equation `ax=b` there are three possible cases:

  1. `a!=0`; root of the equation is `b/a`;
  2. `a=0,b=0`; in this case equation has form `0*x=0` that is true for any x, i.e. root of the equation is any real number.
  3. `a=0,b!=0`; in this case equation has form `0*x=b`, it doesn't have roots.

Example. Solve equation `2/3+x/4+(1-x)/6=(5x)/(12)-1`.

This equation can be transformed into linear. Multiply both sides by 12 (least common multiplier of denominators, 3,4,6,12): `12(2/3+x/4+(1-x)/6)=12((5x)/(12)-1)`.

This can be rewritten as `8+3x+2(1-x)=5x-12` or `8+3x+2-2x=5x-12`, i.e. `8+2+12=5x-3x+2x`. From this we have that `4x=22` or `x=11/2`.