System of Equations
Two or three linear equations with as many unknowns: find the values that make them all true at once. Get exact fractions, the working with Cramer’s rule, a check, and the lines crossing on a graph — or find out why there is no solution, or infinitely many.
What Is a System of Equations?
Each linear equation with two unknowns is a straight line: all the pairs (x, y) that make it true. Solving two of them together means finding the point that lies on both lines — where they cross. With three unknowns, each equation is a flat plane in space, and three planes usually meet in a single point.
Lines can also be parallel (no solution) or lie on top of each other (infinitely many). The determinant D tells them apart: if D is not zero, there is exactly one solution.
Live Graph
linesCramer’s Rule
2 unknowns| Quantity | Formula |
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Three Ways to Solve
Write one unknown in terms of the other from one equation, and put that into the other equation. Handy when one coefficient is 1.
Multiply equations so that one unknown has opposite coefficients, then add them: that unknown cancels out. This is what Gaussian elimination does for bigger systems.
Each unknown is a ratio of two determinants: replace that unknown’s column by the right-hand sides. Neat for 2 or 3 unknowns, and it shows at once when there is no unique solution (D = 0).
Sources & Verification
- G. Strang, Introduction to Linear Algebra, 5th ed., Wellesley-Cambridge Press, 2016 — Ch. 2 (Solving Linear Equations).
- D. C. Lay, S. R. Lay & J. J. McDonald, Linear Algebra and Its Applications, 5th ed., Pearson, 2016 — Ch. 1 (Linear Equations in Linear Algebra).