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Recitation 1A

Today we covered two examples.

Example 1: Solve the given system of equations. x1−2x2+x3=02x2−8x3=0−4x1+5x2+9x3=9.\begin{aligned}x_1 & - & 2x_2 & + & x_3 & = & 0\\& & 2x_2 & - & 8x_3 & = & 0\\-4x_1 & + & 5x_2 & + & 9x_3 & = & 9.\end{aligned}

Solution 1: We turn the system into an augmented matrix. [1−21002−88−4599]\begin{bmatrix}1 & -2 & 1 & 0\\0 & 2 & -8 & 8\\-4 & 5 & 9 & 9\end{bmatrix} Add 4 times the first row to the third row. [1−21002−880−3139]\begin{bmatrix}1 & -2 & 1 & 0\\0 & 2 & -8 & 8\\0 & -3 & 13 & 9\end{bmatrix} Add the second row to the first row. [10−7802−880−3139]\begin{bmatrix}1 & 0 & -7 & 8\\0 & 2 & -8 & 8\\0 & -3 & 13 & 9\end{bmatrix} Divide the second row by 2. [10−7801−440−3139]\begin{bmatrix}1 & 0 & -7 & 8\\0 & 1 & -4 & 4\\0 & -3 & 13 & 9\end{bmatrix} Add 3 times the second row to the third row. [10−7801−4400121]\begin{bmatrix}1 & 0 & -7 & 8\\0 & 1 & -4 & 4\\0 & 0 & 1 & 21\end{bmatrix} Add 7 times the third row to the first row and add 4 times the third row to the second row. [1001550108800121]\begin{bmatrix}1 & 0 & 0 & 155\\0 & 1 & 0 & 88\\0 & 0 & 1 & 21\end{bmatrix} This gives the solution x1=155,x2=88,x3=21.x_1=155, x_2=88, x_3=21.

Example 2: Determine if the following system is consistent. x2−4x3=82x1−3x2+2x3=15x1−8x2+7x3=1.\begin{aligned}& & x_2 & - & 4x_3 & = & 8\\2x_1& - & 3x_2 &+ & 2x_3& = & 1\\5x_1 & - & 8x_2 & + & 7x_3 & = & 1.\end{aligned}

Solution 2: First we write down the augmented matrix. [01−482−3215−871]\begin{bmatrix}0 & 1 & -4 & 8\\2 & -3 & 2 & 1\\5 & -8 & 7 & 1\end{bmatrix} Interchange the first row and the second row because we want to obtain x1x_1 in the first equation. [2−32101−485−871]\begin{bmatrix}2 & -3 & 2 & 1\\0 & 1 & -4 & 8\\5 & -8 & 7 & 1\end{bmatrix} Add -5/2 times row 1 to row 3. [2−32101−480−1/22−3/2]\begin{bmatrix}2 & -3 & 2 & 1\\0 & 1 & -4 & 8\\0 & -1/2 & 2 & -3/2\end{bmatrix} Add 1/2 times row 2 to row 3. [2−32101−480005/2]\begin{bmatrix}2 & -3 & 2 & 1\\0 & 1 & -4 & 8\\0 & 0 & 0 & 5/2\end{bmatrix} The last row in the augmented matrix says 0=5/20=5/2 which is impossible. Therefore the original system is inconsistent, i.e., has no solution.