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Determine the number of solutions each system will have.
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and
The two lines have the same slope, but they have different
y
-intercepts, so they are
parallel lines. Parallel lines do not intersect, so this system has no solution.
So far in this course, most or all of the steps in calculations and algebraic work have been shown. However, once it is clear that an audience can follow a solution with fewer steps, intermediate steps are often excluded.
Compare the steps shown in
Example 1, part b., to the steps shown below for the rearrangement of the equation .
Throughout
Unit 8, some of the intermediate solution steps will not be shown.
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and
Rearrange each equation into slope-intercept form for comparison purposes.
The two lines have different slopes,
so they will have one point of
intersection, and the system has one
solution.
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and
The two lines have the same slope,
but it cannot be determined if they are
parallel or coincident by inspection.
Rearrange the first equation from
slope-point form to slope-intercept
form.
The two equations are equivalent, so the two lines are coincident and the system has an
infinite number of solutions.
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