Linear systems: complete lesson (substitution, combination, matrices)

What is a system?

A linear system is a set of equations in several unknowns to be satisfied all at once.

Example: $\begin{cases} 2x + y = 5 \\ x - y = 1 \end{cases}$

A solution is a pair $(x, y)$ (or triple, etc.) satisfying all equations.

Method 1: substitution

1. Isolate one unknown in one equation.
2. Substitute in the other equation.
3. Solve the single-variable equation.
4. Go back to the first equation to find the other unknown.

Example: from $x - y = 1$, get $x = y + 1$. Substitute in the first: $2(y + 1) + y = 5$ ⇒ $3y = 3$ ⇒ $y = 1$, then $x = 2$.

Method 2: linear combination

1. Multiply each equation to align coefficients of one unknown.
2. Add or subtract to eliminate that unknown.
3. Solve the single-variable equation.
4. Substitute back in an original equation.

Example: adding both equations gives $3x = 6$ ⇒ $x = 2$, then $y = 1$.

Method 3: Gauss elimination (3+ unknowns)

For $n$ equations in $n$ unknowns, transform the system via elementary operations:

• Swap two rows.
• Multiply a row by a nonzero real.
• Add a multiple of one row to another.

Goal: get a triangular system (zeros below the diagonal), then back-substitute.

Number of solutions (2 equations, 2 unknowns)

$\begin{cases} a x + b y = c \\ a' x + b' y = c' \end{cases}$

Determinant: $\Delta = a b' - a' b$.

• $\Delta \neq 0$ ⇒ unique solution (intersecting lines).
• $\Delta = 0$ and system consistent ⇒ infinitely many solutions (coincident lines).
• $\Delta = 0$ and system inconsistent ⇒ no solution (parallel lines).

Matrix form

$\begin{pmatrix} a & b \\ a' & b' \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} c \\ c' \end{pmatrix}$

If $A$ invertible, $X = A^{-1} B$.

Real-life applications

• Mixing problems (2 concentrations).
• Economic optimization (2 products).
• Force systems in physics.

Common mistakes

1. Forgetting to verify in both equations at the end.
2. Sign errors when subtracting equations.
3. Concluding “no solution” too fast (sometimes it’s infinitely many).
4. Winging it instead of applying a systematic method.