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Here is the problem: 4x minus 5y equals 34; 7x plus 9y equals negative 47.

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2 linear equations in 2 unknowns (x, y).

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Write every equation in standard form. Move all variable terms to the left and constants to the right.

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Build the augmented matrix [A | b]. Columns: x, y | constant.

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Gauss–Jordan elimination. Use row operations to make each pivot 1 and clear the rest of its column.

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Divide R1 by 4.

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Clear column 1 using the pivot in R1.

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Divide R2 by 71 over 4.

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Clear column 2 using the pivot in R2.

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Reduced row-echelon form.

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Read off the unique solution.

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So the answer is x equals 1, y equals negative 6.

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The system has exactly one solution; substituting it back satisfies every equation.

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Equation 1: 4x minus 5y equals 34, which checks out.

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Equation 2: 7x plus 9y equals negative 47, which checks out.