Strategy #8: Recognize When a System Has No Solution or Infinite Solutions

Category: Algebra

A system of two linear equations has:

• One solution when the lines intersect (different slopes)

• No solution when the lines are parallel (same slope, different y-intercepts)

• Infinite solutions when the lines are identical (same slope, same y-intercept)

Why it matters: The SAT frequently asks "for what value of k does this system have no solution?" or "infinitely many solutions?" These questions become simple once you recognize the slope condition.

Practice tip: Write both equations in slope-intercept form. Compare slopes and y-intercepts. No solution → same slope, different intercept. Infinite solutions → exactly the same equation.