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 y-intercepts are identical.
Inconsistent: Lines are parallel and never intersect (no solution). Slopes are equal, but y-intercepts differ.

Algebraic Methods

Substitution: Solve one equation for one variable and substitute the resulting expression into the second equation.

Elimination: Multiply equations by scale factors so adding or subtracting cancels one variable symmetrically.

Cramer's Rule (Determinants): Solve using linear matrix determinants:
x = D_x / D | y = D_y / D (D ≠ 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