WEBVTT

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Here is the problem: x plus y equals 3; x plus y equals 4.

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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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Clear column 1 using the pivot in R1.

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Clear column 3 using the pivot in R2.

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Reduced row-echelon form.

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A row reads 0 equals non-zero. Row 2 says 0 equals 1, which is impossible, so the equations contradict each other.

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So the answer is No solution (the system is inconsistent).

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The equations contradict each other, so no values satisfy them all at once.

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The coefficient rows are dependent but the constants are not, which is the signature of parallel lines or planes.