# 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.