The Three Possible Outcomes
This Grade 11 math skill from enVision Algebra 1 explains the three possible outcomes when solving a system of two linear equations: one solution, no solution, or infinitely many solutions. Two lines with different slopes intersect at exactly one point (consistent and independent system). Two lines with the same slope but different y-intercepts are parallel and never intersect (inconsistent system, no solution). Two lines with identical slopes and y-intercepts are the same line (consistent and dependent system, infinite solutions). Students use these classifications to analyze any linear system.
Key Concepts
Property A system of equations can be classified by the number of solutions, which is determined entirely by how the two lines relate to each other visually and algebraically: One Solution (Intersecting): The lines have different slopes. The system is consistent and independent. No Solution (Parallel): The lines have the exact same slope but different y intercepts. The system is inconsistent. Infinite Solutions (Coincident): The lines have the same slope and the same y intercept. The system is consistent and dependent.
Examples One Solution: The lines $y = x + 1$ and $y = x + 3$ intersect at the point $(1, 2)$, giving exactly one solution. No Solution: The lines $y = 2x + 1$ and $y = 2x + 4$ are parallel (same slope, different y intercepts) and never intersect. Infinite Solutions: The equations $y = 3x 2$ and $6x 2y = 4$ represent the same line when graphed, so every point on the line is a solution.
Explanation When you graph two lines, they can only relate in three ways: they cross once, they never cross because they are parallel, or they are actually the exact same line. By simply looking at the 'm' and 'b' in their equations, you can instantly predict how many solutions the system has without even needing to draw the graph!
Common Questions
What are the three possible outcomes for a system of linear equations?
A system of two linear equations has exactly one solution (intersecting lines with different slopes), no solution (parallel lines with same slope but different y-intercepts), or infinitely many solutions (coincident lines — same slope and y-intercept).
When does a system of equations have one solution?
A system has exactly one solution when the two lines have different slopes and intersect at one point. This is called a consistent and independent system.
When does a system of equations have no solution?
A system has no solution when the two lines are parallel — they have the same slope but different y-intercepts. Parallel lines never intersect, so there is no point satisfying both equations. This is an inconsistent system.
When does a system of equations have infinite solutions?
A system has infinite solutions when both equations describe the same line (coincident lines) — they have identical slopes and y-intercepts. Every point on the line satisfies both equations. This is a consistent and dependent system.
What do consistent, inconsistent, and dependent mean for systems?
Consistent means the system has at least one solution. Inconsistent means it has no solution (parallel lines). Independent means one unique solution. Dependent means infinite solutions (same line).