Linear Equations (Two Variables)

Understand systems of two linear equations in a 2D coordinate space. Learn to solve systems using graphing intersections and algebraic determinants (Cramer's Rule).

Algebra Core

Systems of Two Equations

A **system of linear equations** consists of two or more equations with the same variables. Solving the system means finding values for $x$ and $y$ that satisfy both equations simultaneously.

System Classifications

Consistent Independent: Lines intersect at exactly **one point**. The slopes are different.
Consistent Dependent: Lines overlap completely (**infinite solutions**). Slopes and intercepts are identical.
Inconsistent: Lines are parallel and never intersect (**no solution**). Slopes are equal, but intercepts differ.

Algebraic Methods

- **Substitution**: Solve one equation for one variable and substitute the result into the second equation.
- **Elimination**: Multiply one or both equations by constants so that adding them cancels out one variable.
- **Determinants (Cramer's Rule)**: Solve using ratio of coefficients matrices: $x = D_x/D$, $y = D_y/D$ (when determinant $D \neq 0$).

Coordinate Map

Graphing Intersection Solver

Adjust coefficient sliders below to move the two lines. The coordinate system dynamically displays intersections and evaluates algebraic determinants.

Line 1 (Red)

a₁ (x coeff) = 1

b₁ (y coeff) = 1

c₁ (constant) = 3

Line 2 (Blue)

a₂ (x coeff) = -1

b₂ (y coeff) = 1

c₂ (constant) = 1