How do you determine whether a linear system has one solution, many solutions, or no solution when given 4x-6y=2 and 5x+3y= 1?

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How do you determine whether a linear system has one solution, many solutions, or no solution when given 4x-6y=2 and 5x+3y= 1?

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Answer 1 · 2015-10-16T12:39:30

Compare the slopes of the two linear equations.
Since the slopes are not equal there is exactly one solution.

Explanation:

Linear equations with either many solutions or with no solutions must have the same slope (colinear for many solutions; parallel but displaced for no solutions).

$4x-6y=2$ has a slope of $-4/(-6) = 2/3$

$5x+3y=1$ has a slope of $-5/3$

Since the slopes are different the equations are neither colinear nor parallel.

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How do you determine whether a linear system has one solution, many solutions, or no solution when given 4x-6y=2 and 5x+3y= 1?

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