Systems in Three Variables: Practice

A solution gives one value for each variable, and there are three variables.

$(x,\ y,\ z)$

So a solution is three numbers packaged as an ordered triple.

One equation in three unknowns leaves two degrees of freedom: with at least one nonzero coefficient, one variable can be solved from the other two, which are then free.

$z = \frac{d - ax - by}{c} \quad (c \neq 0)$

Two free coordinates sweep out a flat two-dimensional sheet, a plane.

Substitute the three coordinates and add.

$1 + 2 + 3 = 6$

The left side equals the right side, so the triple is a solution.

Substitute the known values into the equation and solve for $x$.

$x + 1 + 2 = 6 \ \Rightarrow \ x = 3$

This is the back-substitution step.

No triple can make $0$ equal a nonzero number, so the reduced statement is impossible.

$0 = 5 \text{ is false} \ \Rightarrow \ \text{no solution}$

The system is inconsistent.

Triangular form drops one leading variable per row: the first equation has all three unknowns, the second has two, the third has one.

$x + y + z = 1, \quad y + z = 2, \quad z = 3$

The other systems keep an $x$ (or a $y$) alive in a lower row, so they are not triangular.

The bottom row gives $z = 4$. Substitute into the middle row.

$y + 4 = 1 \ \Rightarrow \ y = -3$

Back-substitution works upward from the known bottom value.

From the earlier rows, $z = 4$ and $y = -3$. Put both into the top row.

$x + (-3) + 4 = 2 \ \Rightarrow \ x + 1 = 2 \ \Rightarrow \ x = 1$

The full solution is $(1, -3, 4)$.

The system already unlocks from the top down: the first equation hands you $x = 2$, and each later row adds one new unknown. The second gives $y = 3$. Substitute both into the third.

$2 + 3 + z = 9 \ \Rightarrow \ z = 4$

The solution is $(2, 3, 4)$.

If two planes never meet, no point can lie on both of them, let alone on all three.

$\text{no common point} \ \Rightarrow \ \text{no solution}$

The system is inconsistent.

A system is consistent when it has at least one solution and inconsistent when it has none.

$\text{no solution} \ \Rightarrow \ \text{inconsistent}$

"Dependent" describes a consistent system with infinitely many solutions, a different case.

Among consistent systems, one solution is called independent and infinitely many is called dependent.

$\text{consistent} + \text{infinitely many} \ \Rightarrow \ \text{dependent}$

The extra equations depend on the others rather than adding new information.