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In this problem, we are given a system of linear equations that we want to solve.
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We have three equations and three unknown variables, x1, x2, and x3.
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To solve for x1, x2, x3, we want to use the gaussian elimination method.
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To start off, we're going to write our augmented matrix, which corresponds to a matrix on the left -hand side that contains the coefficients of our variables, and on the right -hand side we have a row vector, a column vector, sorry, containing the solutions.
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The goal is to transform this matrix into row echelon form.
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That is, we want a matrix that has null elements in the bottom right -hand triangle.
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To do so, we can do any three of the following transformations.
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We can multiply rows by a constant.
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We could swap rows together or we could add rows amongst themselves after multiplying a scalar.
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So let's start by taking row 2, subtracting it by 2 times row 1 and inserting it back into row 2.
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For now we're not going to manipulate our top row, so i'm just going to re -transcribe that.
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And we'll find in our second row, 0 minus 5, 5, and 25 in our solution matrix.
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Next, we're going to take row 3, and we're going to add 4 times row 1 and insert it back into row 3.
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So we'll find in our bottom row, 0, minus 5, 10, and 45.
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Now we see that we have a factor 5 in common in the second row and third row.
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So let's take these rows and multiply them by constant 1 over 5.
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So we're going to take row 2, divided by 5, and put it back into row 2.
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Same thing.
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We're gonna take, same thing for row three.
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We'll take row three, divided by five, and put it back into row three.
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So we'll find 1, minus 2, 3, 0 minus 1, and 0 minus 1, and 0 minus 1, 2.
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On the left -hand side matrix, and we'll have a right -hand side column vector of 16, 5, and 9 .5, and that.
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Okay, so now let's try to eliminate this one by subtracting it with the second row.
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So let's take row three, subtract it by row two, and put that back into row three.
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We'll find in the last row 0, 0, 0, 1.
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4.
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And the rest remains unchanged...