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

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Solving a System of Linear Equations with Three Variables by Elimination

Example

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

To solve the system $\left\{\begin{array}{l} x + 2y - 3z = -1 \\ x - 3y + z = 1 \\ 2x - y - 2z = 2 \end{array}\right.$ by elimination, first eliminate the variable $z$ . Multiplying the second equation by $3$ and adding it to the first equation yields the two-variable equation $4x - 7y = 2$ . Next, multiply the second equation by $2$ and add it to the third equation to generate another two-variable equation, $4x - 7y = 4$ . This forms a new sub-system: $\left\{\begin{array}{l} 4x - 7y = 2 \\ 4x - 7y = 4 \end{array}\right.$ . To eliminate a variable from this new sub-system, multiply the second equation by $-1$ and add it to the first equation. This completely eliminates the remaining variables, resulting in the mathematically false statement $0 = -2$ . Because we are left with a false statement, the system is explicitly inconsistent and has no solution.

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Updated 2026-04-25

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OpenStax Intermediate Algebra

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