91Ó°ÊÓ

The elimination method is a powerful technique for solving systems of linear equations. It involves adding or subtracting equations to eliminate one of the variables, making it simpler to solve for the remaining variable. Here, we start with two equations in the variables \( c \) and \( n \):

\[ 10c - 16n = 3 \]\[-5c + 8n = 9 \]

We notice that if we multiply the second equation by 2, the coefficients of \( c \) become equal but with opposite signs:

\[-10c + 16n = 18 \]

By adding this to the first equation, the \( c \) terms cancel out:

(10c - 16n) + (-10c + 16n) = 3 + 18

This simplifies to 0 = 21, highlighting the power of the elimination method in signaling inconsistencies when no solutions exist.

A contradiction in equations occurs when we end up with a statement that is always false, like 0 = 21. Let's look at our system again:

\[10c - 16n = 3 \]\[-10c + 16n = 18 \]

When we added these equations, the term for variable \( c \) canceled out:

0 + 0 = 21 or

0 = 21

This result is clearly false and indicates that no combination of \( c \) and \( n \) will satisfy both equations at the same time. Thus, we have encountered a contradiction, meaning there is no solution to the given system of equations.

When dealing with systems of equations, it’s possible to find that there is no solution. This happens if the equations represent parallel lines, which never intersect. In our example, the contradiction 0 = 21 signals that:

• Neither equation can be satisfied by the same \( c \) and \( n \).
• Graphically, if plotted, these lines would be parallel and never meet.

Thus, this system is inconsistent, producing no solution under any circumstance. When using the elimination method, such contradictions immediately indicate no common solution exists.