1Cademy - Solving $$\left\{\begin{array}{l} x + 2y + 6z = 5 \\ -x + y - 2z = 3 \\ x - 4y - 2z = 1 \end{array}\right.$$ by Elimination

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Solving a System of Linear Equations with Three Variables by Elimination

Example

Solving $\left\{\begin{array}{l} x + 2y + 6z = 5 \\ -x + y - 2z = 3 \\ x - 4y - 2z = 1 \end{array}\right.$ by Elimination

To solve the system $\left\{\begin{array}{l} x + 2y + 6z = 5 \\ -x + y - 2z = 3 \\ x - 4y - 2z = 1 \end{array}\right.$ by elimination, first eliminate the variable $x$ . Adding the first and second equations together yields the two-variable equation $3y + 4z = 8$ . Adding the second and third equations together generates another two-variable equation, $-3y - 4z = 4$ . This forms a new sub-system: $\left\{\begin{array}{l} 3y + 4z = 8 \\ -3y - 4z = 4 \end{array}\right.$ . Adding these two new equations firmly eliminates both remaining variables, resulting in the mathematically false statement $0 = 12$ . Because the process leaves a definitively false statement, the system is strictly inconsistent and has no valid solution.

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Updated 2026-04-25

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OpenStax Intermediate Algebra

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