ALGEBRA · CORE CONCEPT

Systems of linear equations

A system asks for values that satisfy several linear equations at the same time. Algebraically, geometrically and in matrix form, the same structure appears.

\[\begin{cases}2x+y=7\\x-y=2\end{cases}\qquad\Rightarrow\qquad (x,y)=(3,1)\]

1. What is a linear system?

A system of \(m\) linear equations in \(n\) unknowns can be written as

\[A\mathbf{x}=\mathbf{b}\]

where \(A\in\mathbb{R}^{m\times n}\) contains the coefficients, \(\mathbf{x}\in\mathbb{R}^n\) the unknowns, and \(\mathbf{b}\in\mathbb{R}^m\) the constants. A solution must satisfy every row equation simultaneously.

2. How many solutions can a system have?

Exactly oneThe equations intersect in one common point. In a square system this corresponds to full rank.
No solutionThe equations are inconsistent; row reduction produces a contradiction such as \(0=1\).
Infinitely manySome equations are dependent, leaving one or more free variables.

The rank criterion is \(\operatorname{rank}(A)=\operatorname{rank}([A\mid b])\) for consistency. If this common rank also equals the number of unknowns, the solution is unique.

3. Main solving methods

SubstitutionIsolate one variable and insert it into another equation. Excellent for small systems.
EliminationAdd suitable multiples of equations so one variable cancels.
Gaussian eliminationApply elementary row operations to the augmented matrix. This scales to large systems.

4. Gaussian elimination — worked example

Solve \(2x+y=7\) and \(x-y=2\).

\[\left[\begin{array}{cc|c}2&1&7\\1&-1&2\end{array}\right]\xrightarrow{R_1\leftrightarrow R_2}\left[\begin{array}{cc|c}1&-1&2\\2&1&7\end{array}\right]\xrightarrow{R_2-2R_1}\left[\begin{array}{cc|c}1&-1&2\\0&3&3\end{array}\right]\]

From the second row, \(y=1\). Substituting into the first row gives \(x=3\).

Elementary row operations preserve the solution set because they replace equations by equivalent linear combinations.

5. Geometric interpretation

In two variables, each linear equation is a line. The solution of the system is their intersection.

unique solution

Parallel distinct lines correspond to no solution; identical lines correspond to infinitely many solutions.

6. Interactive 2×2 solver

Enter coefficients for \(a_1x+b_1y=c_1\) and \(a_2x+b_2y=c_2\).

Enter coefficients and solve.

7. Common mistakes

  • Applying an operation to only one equation side instead of the whole row.
  • Dividing by a pivot without checking whether it is zero.
  • Stopping at row-echelon form without back-substitution.
  • Assuming every system has a unique solution.
  • Confusing an overdetermined system with an inconsistent system — extra equations can still be compatible.

8. Practice

  1. \(x+y=5,\ x-y=1\)
  2. \(2x+4y=6,\ x+2y=3\)
  3. \(x+y=2,\ 2x+2y=5\)
Solutions

1) \((3,2)\). 2) Infinitely many solutions: the equations are equivalent. 3) No solution: the equations contradict each other.