Systems of Linear Equations

What is a System?

Multiple equations, multiple variables. Find values that satisfy all equations at once.

Each equation alone has infinite solutions. Together, they narrow it down.

Example

$x + y = 10$ has many solutions: (5,5), (6,4), (7,3), …

$x - y = 2$ also has many solutions: (3,1), (4,2), (6,4), …

But only (6, 4) satisfies both.

Solving by Elimination

Add or subtract equations to cancel out a variable.

$x + y = 10$ $x - y = 2$ → add them

$2x = 12$ → $x = 6$

Plug back: $6 + y = 10$ → $y = 4$

Solving by Substitution

Solve one equation for one variable, then plug into the other.

From $x + y = 10$, get $y = 10 - x$

Substitute into $x - y = 2$:

$x - (10 - x) = 2$ → $2x - 10 = 2$ → $x = 6$

Plug back: $y = 10 - 6 = 4$