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Homogeneous System of Equations

Homogeneous System of Equations

A system, usually a linear system, in which every constant term is zero.

 

Homogeneous System of Equations: 2x−4y+5z=0, x+7y+z=0, −x+2y+3z=0

Key Formula

{a11x1+a12x2+⋯+a1nxn=0a21x1+a22x2+⋯+a2nxn=0⋮am1x1+am2x2+⋯+amnxn=0\begin{cases} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n = 0 \\ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n = 0 \\ \quad\vdots \\ a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n = 0 \end{cases}
Where:
  • aija_{ij} = The coefficient in the i-th equation for the j-th variable
  • x1,x2,…,xnx_1, x_2, \ldots, x_n = The unknown variables to solve for
  • mm = The number of equations in the system
  • nn = The number of unknowns in the system
  • 00 = The constant term on the right-hand side, always zero in a homogeneous system

Worked Example

Problem: Solve the homogeneous system: 2x + 4y − 2z = 0 x + 2y + z = 0 3x + 6y − z = 0
Step 1: Write the augmented matrix for the system. Since the system is homogeneous, the last column is all zeros.
[24−20121036−10]\left[\begin{array}{ccc|c} 2 & 4 & -2 & 0 \\ 1 & 2 & 1 & 0 \\ 3 & 6 & -1 & 0 \end{array}\right]
Step 2: Swap R1 and R2 to place the leading 1 in the top row, then eliminate the x-terms below it. Replace R2 with R2 − 2R1 and R3 with R3 − 3R1.
[121000−4000−40]\left[\begin{array}{ccc|c} 1 & 2 & 1 & 0 \\ 0 & 0 & -4 & 0 \\ 0 & 0 & -4 & 0 \end{array}\right]
Step 3: Replace R3 with R3 − R2 to eliminate the duplicate row, then divide R2 by −4.
[121000100000]\left[\begin{array}{ccc|c} 1 & 2 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]
Step 4: Back-substitute: from R2, z = 0. Substitute into R1 to get x + 2y + 0 = 0, so x = −2y. The variable y is free.
x=−2t,y=t,z=0(t∈R)x = -2t,\quad y = t,\quad z = 0 \quad (t \in \mathbb{R})
Step 5: Write the general solution in vector form.
(xyz)=t(−210)\begin{pmatrix} x \\ y \\ z \end{pmatrix} = t\begin{pmatrix} -2 \\ 1 \\ 0 \end{pmatrix}
Answer: The system has infinitely many solutions: (x, y, z) = t(−2, 1, 0) for any real number t. The trivial solution t = 0 gives (0, 0, 0).

Another Example

This simpler 2-variable example shows that when equations are dependent (one is a multiple of the other), a homogeneous system always yields a nontrivial solution with a free variable, not just the trivial solution.

Problem: Solve the homogeneous system: x + 3y = 0 2x + 6y = 0
Step 1: Write the augmented matrix.
[130260]\left[\begin{array}{cc|c} 1 & 3 & 0 \\ 2 & 6 & 0 \end{array}\right]
Step 2: Replace R2 with R2 − 2R1.
[130000]\left[\begin{array}{cc|c} 1 & 3 & 0 \\ 0 & 0 & 0 \end{array}\right]
Step 3: The second row is all zeros, so y is a free variable. From R1: x = −3y. Let y = t.
x=−3t,y=t(t∈R)x = -3t,\quad y = t \quad (t \in \mathbb{R})
Answer: The solution set is (x, y) = t(−3, 1) for any real number t. The two equations are multiples of each other, so the system has infinitely many solutions along a line through the origin.

Frequently Asked Questions

Does a homogeneous system always have a solution?
Yes, always. Setting every variable to zero satisfies every equation because each right-hand side is already zero. This all-zeros answer is called the trivial solution. The real question is whether nontrivial solutions (where at least one variable is nonzero) also exist.
When does a homogeneous system have infinitely many solutions?
A homogeneous system has infinitely many (nontrivial) solutions whenever the number of unknowns exceeds the number of independent equations — that is, when there is at least one free variable after row reduction. In particular, if a system has more unknowns than equations, it is guaranteed to have nontrivial solutions.
What is the difference between the trivial and nontrivial solution?
The trivial solution assigns zero to every variable: x₁ = x₂ = … = xₙ = 0. A nontrivial solution is any solution in which at least one variable is nonzero. For many applications, finding nontrivial solutions is the main goal.

Homogeneous System vs. Non-Homogeneous System

Homogeneous SystemNon-Homogeneous System
DefinitionAll constant terms on the right-hand side are zeroAt least one constant term on the right-hand side is nonzero
General formAx = 0Ax = b, where b ≠ 0
Guaranteed solution?Always has the trivial solution (0, 0, …, 0)May have no solution at all
Solution structureSolution set forms a subspace (passes through the origin)Solution set is a translated copy of the homogeneous solution set
Example2x + 3y = 02x + 3y = 5

Why It Matters

Homogeneous systems appear throughout algebra and applied mathematics. In linear algebra courses, they are central to understanding null spaces, eigenvalue problems, and linear independence of vectors. In physics and engineering, homogeneous systems model equilibrium states and vibration modes where external forces are absent.

Common Mistakes

Mistake: Concluding that the only solution is the trivial solution without performing row reduction.
Correction: You must row-reduce the coefficient matrix to check for free variables. If any free variable exists, nontrivial solutions exist. A system with more unknowns than equations always has nontrivial solutions.
Mistake: Forgetting that the solution set of a homogeneous system must include the zero vector.
Correction: Every homogeneous system has (0, 0, …, 0) as a solution. If your answer does not include this point (for example, when the parameter equals zero), recheck your algebra — something went wrong.

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