When you solve a system of equations by either the substitution or the elimination (addition) method, how do you determine whether the system of equations is independent, dependent, or inconsistent?

Briefly Speaking.

System of equations are independent, if 2nd equation is not multiplication of first equation by some number .

Example

X + 3y = 5 ( 1)

2 ( x +5 ) - y = 20 -7y

Simplify the 2nd equation:

2 X + 10 - y = 20 -7y

2 X + 6y = 10 ( #3)

Equation ( # 3) is the (1) multiplied by 2, and every (X , Y ) value that works in first will work in

2nd as well. So the system is dependent.

System is inconsistent when 2 lines are parallel ( have same slope) and dont have a common point

of intersect.

3X + 5Y = 15

6X + 10Y =21

5Y = -3X + 15 Y = -3/5 X +3

6 X + 10Y = 21

10y = -6X +21 Y = -3/5 X + 21/ 10

This System is inconsistent you see that 2 lines have same slope, and are parallel.