Lesson 4: Number of Solutions and Special Cases

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!

Property

When using elimination, if both variables are eliminated, the resulting mathematical statement determines the number of solutions:

• If the resulting equation is a false statement (e.g., $0 = c$), the system has no solution.
• If the resulting equation is a true statement (e.g., $0 = 0$), the system has infinitely many solutions.

You can always verify your algebraic solution by graphing the two lines to confirm they intersect exactly at your calculated $(x, y)$ coordinate.

Examples

• No Solution: For the system $2x - y = 5$ and $-4x + 2y = -12$, multiply the first equation by 2 to get $4x - 2y = 10$. Adding this to the second equation results in $0 = -2$, a false statement, meaning there is no solution.
• Infinitely Many Solutions: For the system $x + 3y = 6$ and $2x + 6y = 12$, multiply the first equation by -2 to get $-2x - 6y = -12$. Adding this to the second equation results in $0 = 0$, a true statement, meaning there are infinitely many solutions.
• Graphical Check: If elimination gives $(4, -1)$ as a solution, both original equations should pass exactly through the point $(4, -1)$ when graphed.

Explanation

Sometimes when you perform elimination, both variables cancel out. If you are left with a false statement like $0 = -2$, it means there is no pair of $(x, y)$ that can satisfy both equations because the lines are perfectly parallel. If you are left with a true statement like $0 = 0$, it means the two equations describe the exact same line, resulting in infinitely many solutions.