Solving an Inconsistent System

Example

Solve the system:

$3x-2y=4$

$-6x+47=7$

Solution

To eliminate the variable x, multiply both sides of the first equation by 2 and add the result to the second equation.

$6x-4y=8$

$-6x+4y=7$

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$0=15$

This last equation is clearly false.

Since 0 = 15 is false, the system is inconsistent and has no solution. As suggested by the graph, this means that the graphs of the equations of the system never intersect. (The lines are parallel.) The solution set is $\varnothing$. Here is a graph of this system showing that the equations graphs are parallel lines.