WEBVTT

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Here is the problem: x plus y equals 3; 2x plus 2y equals 6.

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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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Reduced row-echelon form.

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y is a free variable. There are 1 pivot for 2 unknowns, so 1 variable is free. Let y equals t and solve the pivot rows for the rest.

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So the answer is Infinitely many solutions: x equals 3 minus t, y equals t (t any real number).

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The equations are dependent, so they describe the same line/plane (or an intersection that is a whole line), giving infinitely many solutions.

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Rank 1 less than 2 unknowns with no contradictory row, so the solution set is a 1-parameter family.