Mathematics.

systems of equations

Systems of Equations in Three Variables

Algebra II20 minDifficulty5 out of 10

You should know: systems of linear equations

Overview

A system of equations in three variables (x, y, z) is solved by systematically eliminating variables until a single equation in one unknown remains, then back-substituting. Geometrically, each linear equation represents a plane in 3D space, and a unique solution corresponds to the single point where all three planes intersect.

Formal Definition

Definition

A general linear system in three variables:

{a1x+b1y+c1z=d1a2x+b2y+c2z=d2a3x+b3y+c3z=d3\begin{cases} a_1x+b_1y+c_1z = d_1 \\ a_2x+b_2y+c_2z = d_2 \\ a_3x+b_3y+c_3z = d_3 \end{cases}
General form

Properties

Unique solution

Three planes meet at exactly one point  ⟺  the system has a unique solution\text{Three planes meet at exactly one point} \iff \text{the system has a unique solution}

No solution

The planes have no common point (e.g. parallel or forming a triangular prism)  ⟺  inconsistent system\text{The planes have no common point (e.g. parallel or forming a triangular prism)} \iff \text{inconsistent system}

Infinite solutions

The planes share a common line or coincide  ⟺  dependent system\text{The planes share a common line or coincide} \iff \text{dependent system}

Worked Examples

  1. Add equations 1 and 2 to eliminate y.

    (x+y+z)+(2x−y+z)=6+3  ⟹  3x+2z=9(x+y+z)+(2x-y+z) = 6+3 \implies 3x+2z=9

  2. Add equations 1 and 3, doubling equation 1 first, to eliminate y a second (independent) way.

    2(x+y+z)+(x+2y−z)=2(6)+2  ⟹  3x+4y+z=14, instead pair eq.2 and eq.3: (2x−y+z)+(x+2y−z)=3+2  ⟹  3x+y=52(x+y+z)+(x+2y-z) = 2(6)+2 \implies 3x+4y+z=14 \text{, instead pair eq.2 and eq.3: } (2x-y+z)+(x+2y-z)=3+2 \implies 3x+y=5

  3. From eq.1: y = 6-x-z. Substitute into 3x+y=5 to relate x and z, then combine with 3x+2z=9.

    3x+(6−x−z)=5  ⟹  2x−z=−1  ⟹  z=2x+13x+(6-x-z)=5 \implies 2x-z=-1 \implies z=2x+1

  4. Substitute z=2x+1 into 3x+2z=9 and solve for x, then back-substitute for z and y.

    3x+2(2x+1)=9  ⟹  7x=7  ⟹  x=1;z=2(1)+1=3;y=6−1−3=23x+2(2x+1)=9 \implies 7x=7 \implies x=1;\quad z=2(1)+1=3;\quad y=6-1-3=2

Answer: x = 1, y = 2, z = 3

Practice Problems

Difficulty 6/10

Solve: x+y+z=4, 2x-y+z=8, x+2y-z=-3.

Common Mistakes

Common Mistake

Eliminating the same variable using the same pair of equations twice, producing no new information.

To solve a 3-variable system you need two INDEPENDENT eliminations of the same variable, using different pairs of equations (e.g. eliminate y from eq.1&2, then separately from eq.1&3), producing two new equations in the remaining two variables.

Summary

  • A 3-variable linear system is solved by eliminating one variable at a time until a single-variable equation remains.
  • Geometrically, each equation is a plane; a unique solution is the single point common to all three planes.
  • Use two independent eliminations (different equation pairs) to reduce to a 2-variable system, solve that, then back-substitute.
  • As with 2-variable systems, a 3-variable system can have zero, one, or infinitely many solutions.

References

  1. WebsiteWikipedia — System of linear equations