Systems: No Solution vs Infinite Solutions

Three possible outcomes for a system of two linear equations

One solution (most common): Lines have different slopes → they cross at exactly one point. This system is consistent and independent.

No solution: Lines have the same slope but different y-intercepts → they are parallel and never meet. This system is inconsistent. Example: $y = 2x + 1$ and $y = 2x + 4$ — same slope (2), different intercepts.

Infinite solutions: Lines have the same slope AND same y-intercept → they are the same line. Every point on the line is a solution. This system is dependent. Example: $y = 2x + 1$ and $2y = 4x + 2$ (which simplifies to the same equation).

How to identify from the equations

Compare slopes $m_1, m_2$ and intercepts $b_1, b_2$:

• $m_1 \neq m_2$ → one solution
• $m_1 = m_2$ and $b_1 \neq b_2$ → no solution (parallel)
• $m_1 = m_2$ and $b_1 = b_2$ → infinite solutions (same line)

Try it in the visualization

Adjust slopes and intercepts. When both slopes match but intercepts differ, the lines become parallel and the classification changes to "NO SOLUTION." Make everything match to see "INFINITE SOLUTIONS."