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Mathematics

1 answer:

nikklg [1K]2 years ago

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Explanation:

The solution set for a system of equation is the set of points where the graphs of the equations intersect.

<h3>general case</h3>

A system will have <em>one solution</em> if there is a <em>single point of intersection</em> of the graphs of the equations.

A system will have <em>no solutions</em> if the graphs have <em>no points of intersection</em>.

A system will have an <em>infinite</em> number of <em>solutions</em> if the graphs <em>intersect at an infinite number of points</em>.

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<h3>linear equations</h3>

When the equations are linear equations, their graphs are straight lines. If the lines have different slopes, they must intersect at exactly one point: there will be one solution.

If the lines have the same slope, there are two possibilities:

the lines are parallel -- no solutions
the lines are coincident -- infinite solutions

The attached graph illustrates these cases.

the red and blue lines are the graphs of a system of equations with one solution. Those lines have different slopes
the blue and green lines are the graphs of a system of equations with no solution. Those lines are parallel.
The red and (dotted) purple lines are the graphs of a system of equations with infinite solutions. Those lines are coincident.