Classifying Without Graphing
Determine the number of solutions and classify the system without graphing, by comparing slopes and -intercepts.
Second equation: is already in slope-intercept form.
First equation: Convert to slope-intercept form by isolating :
Compare slopes and y-intercepts:
- First line: ,
- Second line: ,
Because both the slopes and the -intercepts are identical, the two equations describe the same line — the lines are coincident.
The system has infinitely many solutions and is consistent and dependent.
This example complements cases where the slopes differ (one solution, consistent and independent) and where the slopes match but the -intercepts differ (no solution, inconsistent and independent). Here, matching both slope and -intercept confirms that every point on the shared line satisfies both equations.
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Ch.5 Systems of Linear Equations - Elementary Algebra @ OpenStax
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Consistent System of Equations
Inconsistent System of Equations
Independent Equations in a Linear System
Dependent Equations in a Linear System
Classifying Without Graphing
Classifying Without Graphing
Classifying Without Graphing
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