Linear Combination Calculator

A system of linear equations is a set of two or more equations where each variable appears only in the first power — no squares, cubes, roots, or denominators. A solution to the system is a set of values that satisfies all equations simultaneously.

The linear combination method (also called the elimination method) involves multiplying one or both equations by constants so that the coefficients of one variable become equal and opposite. Adding the equations then eliminates that variable, leaving a single equation to solve directly.

First, choose a variable to eliminate. Multiply each equation by a suitable multiplier so the chosen variable's coefficients cancel when the equations are added. Solve the resulting single-variable equation, then substitute back to find the other variable.

Multiply the first equation by 6 to get 30x + 12y = 72, then subtract the second equation (8x + 12y = 28) to get 22x = 44, so x = 2. Substituting back gives 2y = 12 − 10 = 2, so y = 1. The solution is x = 2, y = 1.

The determinant D = a1·b2 − a2·b1. If D ≠ 0 the system has exactly one unique solution. If D = 0 and the numerators are also zero the system is dependent (infinite solutions). If D = 0 but a numerator is non-zero the system is inconsistent (no solution).

A linear combination of vectors is an expression formed by multiplying each vector by a scalar and adding the results — for example, c₁v₁ + c₂v₂. When the only combination that gives the zero vector uses all-zero scalars, the vectors are linearly independent.

Yes. Simply enter negative values for any coefficient or constant. The elimination algorithm works identically regardless of sign, and the calculator will display the correct x and y values along with a step-by-step table.

If the determinant is zero, the two equations are either parallel (no solution — inconsistent) or identical/multiples of each other (infinite solutions — dependent). The calculator detects this automatically and reports the system type in the results.