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is it consistant, inconsistant, or depent

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Tamara J. | Math Tutoring - Algebra and Calculus (all levels)Math Tutoring - Algebra and Calculus (al...
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Notice that this is a system of linear equations, for which we can solve for as follows:

      2x + 5y = -21   ==>   2ยท(2x + 5y = -21)   ==>   4x + 10y = -42

     -4x - 10y = 42            ==>        ==>        ==>  -4x - 10y = 42

Combining these two equations we arrive at the following:

          4x + 10y = -42

         -4x - 10y = 42


   4x - 4x + 10y - 10y = -42 + 42     ==>     0 = 0

When you arrive at an answer that is an identity (e.g., 0 = 0), this indicates that the equations are identical or equivalent. This means that the lines are the same and they intersect at infinitely many points; that is, there are infinitely many solutions to this system of equations. 

When a system of equations has no solutions, then the system in inconsistent. If it has one or more solutions (i.e., infinitely many solutions), then it is consistent. When the solution to one equation in the system is the solution to the other equation in the system, then the equations are identical and the system is dependent. If the system has one unique solution, then the system is independent

Since we've found that the system has infinitely many solutions and the equations are identical, then this system of equations is consistent and dependent.