Special Cases and Graphical Verification
When the elimination method produces a result where both variables cancel, the type of statement that remains determines the number of solutions for the system. In Grade 11 enVision Algebra 1 (Chapter 4: Systems of Linear Equations and Inequalities), a false statement like 0 = −2 signals no solution (parallel lines), while a true statement like 0 = 0 signals infinitely many solutions (the same line). Graphical verification confirms algebraic results by checking whether two lines intersect at the calculated point.
Key Concepts
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.
Common Questions
What happens when both variables cancel during elimination?
If a false statement results (like 0 = −2), the system has no solution. If a true statement results (like 0 = 0), the system has infinitely many solutions.
What does 0 = −2 tell you about a system of equations?
It is a contradiction — no ordered pair satisfies both equations. The lines are parallel and never intersect.
What does 0 = 0 tell you about a system of equations?
It is always true — the two equations represent the same line, so every point on the line is a solution (infinitely many solutions).
How does graphical verification work for a system of equations?
Graph both equations and check whether the lines intersect at the ordered pair you calculated. The intersection point should match your algebraic solution.
How can you verify a solution algebraically?
Substitute the calculated (x, y) values into both original equations and confirm that both equations are satisfied.
What is the graphical interpretation of a system with infinitely many solutions?
Both equations graph as the same line — they are coincident, meaning every point on the line satisfies both equations.