1Cademy - Solving $$\left\{\begin{array}{l} 2x - 2y + 3z = 6 \\ 4x - 3y + 2z = 0 \\ -2x + 3y - 7z = 1 \end{array}\right.$$ by Elimination

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Solving a System of Linear Equations with Three Variables by Elimination

Example

Solving $\left\{\begin{array}{l} 2x - 2y + 3z = 6 \\ 4x - 3y + 2z = 0 \\ -2x + 3y - 7z = 1 \end{array}\right.$ by Elimination

To solve the system $\left\{\begin{array}{l} 2x - 2y + 3z = 6 \\ 4x - 3y + 2z = 0 \\ -2x + 3y - 7z = 1 \end{array}\right.$ by elimination, first eliminate the variable $x$ . Adding the first and third equations together directly yields the two-variable equation $y - 4z = 7$ . Next, multiply the third equation by $2$ and add it to the second equation to produce another two-variable equation, $3y - 12z = 2$ . This forms a new sub-system of equations: $\left\{\begin{array}{l} y - 4z = 7 \\ 3y - 12z = 2 \end{array}\right.$ . To eliminate a variable from this sub-system, multiply the first equation by $-3$ and add it to the second equation. This completely removes the variables and results clearly in the false mathematical statement $0 = -19$ . Because we are left firmly with a false numerical statement, the system is inconsistent and yields no solution.

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Updated 2026-04-25

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OpenStax Intermediate Algebra

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