Systems with No Solutions

Figure 11.3 illustrates two lines with the same slope. They are parallel to each other, so they never intersect. The equations as given read this way:

Figure 11.3. Lines that do not intersect have no solution.

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If you consider the two equations that generate these lines, you end up with a system that has this appearance:

x - y = -3

x - y = 3

If you try to arrive at a solution for this system of equations, you might proceed by multiplying by -1 so that you can eliminate the x variable. Your work proceeds along the following lines:

x - y = -3

-x + y = 3

Given this result, you can then add -x + y = 3 to the first equation:

The addition operation produces an equation that is inconsistent because 0 is not equal to 6. As it is, when you attempt to find values that allow you to create a consistent addition product, your efforts fail. The system does not allow the lines to intersect, so no solution exists.