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When you solve a system of equations by either the substitution or the elimination (addition) method, how do you determine whether the system of equati.....

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4 Answers

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.
 
                 
 
  
           
Suppose I gave you an system of equations that have the same slope
 
y = 2x - 7
2y = 4x + 16
 
y - 2x = -7
2y - 4x = 16
 
Multiply first equation by -2
-2y + 4x = -14
 2y -  4x =   16
 
Add the two equations
 
0 = 2 which is not true.  Therefore the system is inconsistent.
 
Now, let's try another pair of lines
 
  y = 4x + 13
3y = 12x + 39
 
Let's divide the second equation by 3
 
y = 4x + 13 which is precisely the same as the first equation.  In this case the lines are identical and the system is said to be dependent.
 
Finally, if a pair of lines intersect at one point, the system is said to be independent.
When you solve a system of equations, you are looking for a solution that satisfies both equations.  
Each equation can be represented by a line on a coordinate plane.
 
If you solve the equations algebraically and you get an answer which is untrue or impossible such as 0=12, then the two lines will be parallel and the system is inconsistent.
 
If you solve the equations algebraically and you get a solution such as 5=5, then the lines are actually the same line and the system is dependent.
 
If you solve the equations algebraically and you get an answer which is an ordered pair, then the two lines intersect and the system is independent.
 
 
If you use either method and end up with a statement that's always true, like 0 = 0 or 1 = 1, the system of equations is dependent. Here's a simple example to try yourself:
x+y = 2
2x+ 2y = 4.
 
If you use either method and end up with a statement that's always false, like 0 = 1, the system of equations is inconsistent. Here's a simple example:
x + y = 2
2x + 2y = 3
If you use either method and end up with particular numerical values of x and y, the system of equations is independent. Most systems of equations that you've solved in the past are independent.
 
There are a huge number of other methods you could use that apply matrix algebra or graphing methods, but the above technique is useful to remember for algebra courses because it can help to explain what's happening when you're not expecting to find a dependent or inconsistent system. It can be confusing, especially with word problems, when you're trying to find numerical values for variables and instead you discover that all the variables cancel each other and you've obtained the unusual statements that zero equals zero or that zero equals one.