AHL 1.16 (HL)—Systems of linear equations

Classify and solve a linear system.

Row reduction or technology reveals whether equations meet at one point, share a line/plane, or conflict; classify before reporting a solution.

Worked example
x+y=2 and 2x+2y=4 have infinitely many solutions because the equations are dependent.

Worked example
What signals no solution? a contradictory row such as 0=1.

Common boundary
Three equations do not guarantee a unique solution.

Three-equation example: solve $x+y+z=6$, $2x-y+z=3$, and $x+2y-z=2$. Subtracting the first equation from the second gives $x-2y=-3$; subtracting it from the third gives $y-2z=-4$. Hence $x=2y-3$, $y=2z-4$, and substitution into the first equation gives $7z=21$. Therefore $(x,y,z)=(1,2,3)$, a unique solution; substitution checks all three equations.