Linear Combination Calculator

A linear combination calculator does two jobs that share one name. In linear algebra it computes c1v1 + c2v2 + … + cnvn: each vector scaled by its coefficient and added component by component, with the result, its magnitude and every scaling step shown. In algebra class, the linear combination method means solving two equations by scaling them so one variable cancels. This page carries both as tabs, because searchers using the same phrase want both.

Quick Answer

How to use the two tabs

Each tab is a separate tool with its own inputs, results and reset.

1. Vectors: set the number of vectors first (1 to 8), because changing it rebuilds the rows and clears what you typed. Then, per vector, enter its coefficient, its dimension (1 to 10 components), and the components. The tab opens pre-filled with the worked example.
2. Blanks count as zero in the vector tab: an empty component is a 0, and an empty coefficient is a 0. The one refusal is every coefficient at zero.
3. Solve 2×2 system: type the six numbers of a1x + b1y = c1 and a2x + b2y = c2. All six are required; a blank field gets “Enter all six coefficients.” The pre-filled system is the worked example, x = 2 and y = 1.
4. Read the steps, then the check. Both tabs verify the result against the original inputs, and both have Copy and Share buttons that send the full steps with the answer.
5. Reset restores the worked examples on both tabs rather than blanking the fields.

What a linear combination is

A linear combination scales each vector by a number, its coefficient, and adds the scaled vectors component by component: c1v1 + c2v2 + … + cnvn. The set of every result reachable by varying the coefficients is the span of the vectors.

The idea is the backbone of linear algebra: a point reached by walking 2 units of one vector and back 1 unit of another, a colour mixed from weighted channels, a portfolio return built from weighted assets. The Georgia Institute of Technology’s Interactive Linear Algebra chapter on spans is the standard undergraduate framing, and Gilbert Strang’s MIT OpenCourseWare 18.06 linear algebra course builds the whole subject on combinations of columns. Two non-parallel vectors in the plane span the whole plane; two multiples of each other span only a line, which is why the tool flags the two-vector case that collapses to the zero vector as linearly dependent.

Reading the vector steps, padding and the magnitude

The result line gives the combined vector, the line under it restates the combination and the magnitude, and the steps box shows one scaled vector per line so each multiplication can be checked on its own.

Padding. Vectors of different dimensions combine by padding the shorter ones with zeros up to the longest dimension: [3, 4] combined with [1, 1, 1] treats the first as [3, 4, 0]. The steps print the padded form, so the arithmetic you see is the arithmetic that happened. Precision. Numbers print to six decimal places with trailing zeros stripped, so 6.708204 is exact as shown while 2.5 stays 2.5. Magnitude. The magnitude is the length of the result, the square root of the sum of squared components: for [−2, 4, 5] that is √(4 + 16 + 25) = √45 = 6.708204. Magnitude says how far the result sits from the origin; the angle between two vectors is a different question, answered by the dot product calculator, and in three dimensions the perpendicular vector to two inputs comes from the cross product calculator.

One scope note on the dependence flag: it fires only when exactly two vectors combine to the zero vector with non-zero coefficients. With three or more vectors a zero result hints at dependence too, but pinning it down needs a rank check this tab does not run, so it stays silent rather than guess.

The linear combination method for 2×2 systems

The method scales one or both equations so a variable’s coefficients become opposites, adds the equations to eliminate that variable, solves the single-variable equation left behind, and substitutes back for the other variable.

This solver eliminates x first. When both x coefficients are whole numbers it picks multipliers from their least common multiple, the classroom route: for 5 and 8 the LCM is 40, so the equations scale by 8 and −5. With decimal coefficients it scales by the other equation’s x coefficient instead. The determinant check comes first for a reason: if a1b2 − a2b1 is zero there is no unique pair to find, and the solver says so instead of dividing by zero. It does not separate the two zero-determinant cases, parallel lines (no solution) from the same line twice (infinitely many); the linear systems calculator makes that distinction, while this tab’s job is the elimination working itself. The matrix route to the same answer is Cramer’s rule, determinant ratios, which the inverse matrix calculator for 2×2 systems supports from the coefficient matrix side.

The zero-coefficient case deserves a sentence of its own. If the first equation has no x term (a1 = 0), there is nothing to eliminate in it, so the solver solves that equation for y directly and substitutes into the second equation: 0x + 2y = 6 with 3x + y = 9 gives y = 3 from the first equation, then x = 2. The same applies when the second equation is the one missing x.

Worked examples, exactly as the tool prints them

Both pre-filled cases, with the tool’s own steps. Change any number and the same machinery reruns.

Vectors. Compute 2[1, 2, 3] + (−1)[4, 0, 1]:

System. Solve 5x + 2y = 12 and 8x + 12y = 28:

Notice the step text keeps the tool’s raw signs, writing “-40x + -60y” rather than tidying it to “-40x − 60y”. The printed steps are a transcript of the scaling, sign included, so a student copying the working sees the same line the calculator produced.

How this page differs from the other linear calculators

Six pages on this site carry “linear” in the name, and they compute six different things. This one combines vectors and eliminates variables in a 2×2 system; the others do neither.

The honest overlap is with linear systems: both pages accept the same six coefficients. The difference is the deliverable. That page returns the solution and the determinant working; this page returns the elimination transcript a teacher asks to see.

What competing calculators miss

Search results for this phrase split into two camps that rarely acknowledge each other:

• Omni Calculator: the strongest rival, covering the elimination method with worked steps and a vector FAQ. Gap: its vector treatment stays at explanation level beside the system solver; there is no general n-vector combiner with per-vector dimensions on the page.
• OnlyCalculators: runs a dedicated linear combination system solver with step-by-step elimination and no-solution detection, and separately a vector addition tool. Gap: the two halves live on different pages, and its vector side handles plain addition with optional scalar toggles rather than a general coefficient-per-vector combination.
• CalculatorsHub: a clean two-vector combiner with the formula taught properly. Gap: hard-capped at two vectors in two dimensions, with no magnitude, no steps and no system solver.
• Made Calculators: two vectors, two dimensions, and by its own admission no input validation. A malformed entry fails silently there; here a zero-coefficient set and a missing system field both get explicit messages.
• CalcWorkshop and the homework-answer sites: good teaching on the elimination method, including when to pick it over substitution or graphing. Gap: no working tool at all; the examples are fixed before you arrive.
• Calculator.now’s elimination calculator: a solid 2×2 solver that classifies unique, none and infinite solutions. Gap: systems only; the vector half of the search intent goes unserved.
• The gap this page fills: both intents on one page, up to 8 vectors in up to 10 dimensions with scaling steps and magnitude, and an elimination solver that prints its LCM multipliers, scaled equations and final check.

Honest limits of this calculator

• Caps: 8 vectors at most, 10 components per vector at most. The fields clamp typed values back into range when rows are built.
• Zero-determinant systems are not classified. A zero determinant produces one message covering both parallel and identical lines. The answer is still honest (no unique solution), but the page does not tell you which of the two you have.
• The dependence note is narrow. It fires only for exactly two vectors producing the zero vector, as the steps section explains.
• Changing the vector count wipes the rows. Set the count before typing components, or copy your numbers first.
• Systems stop at 2×2. Three equations in three unknowns need a matrix method, not this tab.

Frequently asked questions

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References