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Math  /  Algebra

QuestionSolve using Gauss-Jordan elimination. x1−x2+x3+2x4=−1.6−3x1+5x2+4x3−14x4=13.74x1−2x2+9x3+5x4=−2.13x1−2x2+x3+10x4=−8.4\begin{array}{rr} x_{1}-x_{2}+x_{3}+2 x_{4}= & -1.6 \\ -3 x_{1}+5 x_{2}+4 x_{3}-14 x_{4}= & 13.7 \\ 4 x_{1}-2 x_{2}+9 x_{3}+5 x_{4}= & -2.1 \\ 3 x_{1}-2 x_{2}+x_{3}+10 x_{4}= & -8.4 \end{array}
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Studdy Solution

STEP 1

1. We are given a system of linear equations in the form of an augmented matrix.
2. We will use the Gauss-Jordan elimination method to solve this system.
3. The goal is to transform the matrix to its reduced row echelon form (RREF).

STEP 2

1. Form the augmented matrix for the given system of equations.
2. Use row operations to get a leading 1 (pivot) in the first row, first column.
3. Use row operations to make all other entries in the first column zero.
4. Repeat steps 2-3 for the next columns, ensuring that each leading 1 is to the right of the previous one and that all entries above and below the leading 1 are zero.
5. Continue until the matrix is in reduced row echelon form (RREF).
6. Read off the solutions from the RREF matrix.

STEP 3

Form the augmented matrix for the given system of equations.
[1−112−1.6−354−1413.74−295−2.13−2110−8.4] \left[\begin{array}{cccc|c} 1 & -1 & 1 & 2 & -1.6 \\ -3 & 5 & 4 & -14 & 13.7 \\ 4 & -2 & 9 & 5 & -2.1 \\ 3 & -2 & 1 & 10 & -8.4 \end{array}\right]

STEP 4

Ensure the leading coefficient (pivot) in the first row is 1. It is already 1.

STEP 5

Use row operations to make all other entries in the first column zero. - Multiply the first row by 3 and add it to the second row. - Multiply the first row by -4 and add it to the third row. - Multiply the first row by -3 and add it to the fourth row.
R2=R2+3R1R3=R3−4R1R4=R4−3R1 R_2 = R_2 + 3R_1 \\ R_3 = R_3 - 4R_1 \\ R_4 = R_4 - 3R_1
[1−112−1.6027−88.9025−34.301−24−3.6] \left[\begin{array}{cccc|c} 1 & -1 & 1 & 2 & -1.6 \\ 0 & 2 & 7 & -8 & 8.9 \\ 0 & 2 & 5 & -3 & 4.3 \\ 0 & 1 & -2 & 4 & -3.6 \end{array}\right]

STEP 6

Next, ensure the leading coefficient (pivot) in the second row is 1. Currently, it is 2, so divide the second row by 2.
R2=12R2 R_2 = \frac{1}{2} R_2
[1−112−1.60172−44.45025−34.301−24−3.6] \left[\begin{array}{cccc|c} 1 & -1 & 1 & 2 & -1.6 \\ 0 & 1 & \frac{7}{2} & -4 & 4.45 \\ 0 & 2 & 5 & -3 & 4.3 \\ 0 & 1 & -2 & 4 & -3.6 \end{array}\right]

STEP 7

Use row operations to make all other entries in the second column zero. - Subtract twice the second row from the third row. - Subtract the second row from the fourth row. - Add the second row to the first row.
R3=R3−2R2R4=R4−R2R1=R1+R2 R_3 = R_3 - 2R_2 \\ R_4 = R_4 - R_2 \\ R_1 = R_1 + R_2
[1092−22.850172−44.4500−25−4.600−1128−8.05] \left[\begin{array}{cccc|c} 1 & 0 & \frac{9}{2} & -2 & 2.85 \\ 0 & 1 & \frac{7}{2} & -4 & 4.45 \\ 0 & 0 & -2 & 5 & -4.6 \\ 0 & 0 & -\frac{11}{2} & 8 & -8.05 \end{array}\right]

STEP 8

Next, ensure the leading coefficient (pivot) in the third row is 1. Currently, it is -2, so divide the third row by -2.
R3=−12R3 R_3 = -\frac{1}{2} R_3
[1092−22.850172−44.45001−522.300−1128−8.05] \left[\begin{array}{cccc|c} 1 & 0 & \frac{9}{2} & -2 & 2.85 \\ 0 & 1 & \frac{7}{2} & -4 & 4.45 \\ 0 & 0 & 1 & -\frac{5}{2} & 2.3 \\ 0 & 0 & -\frac{11}{2} & 8 & -8.05 \end{array}\right]

STEP 9

Use row operations to make all other entries in the third column zero. - Add 112\frac{11}{2} times the third row to the fourth row. - Subtract 92\frac{9}{2} times the third row from the first row. - Subtract 72\frac{7}{2} times the third row from the second row.
R4=R4+112R3R1=R1−92R3R2=R2−72R3 R_4 = R_4 + \frac{11}{2} R_3 \\ R_1 = R_1 - \frac{9}{2} R_3 \\ R_2 = R_2 - \frac{7}{2} R_3
[100132−17.55010−92−3.65001−522.3000−1524.6] \left[\begin{array}{cccc|c} 1 & 0 & 0 & \frac{13}{2} & -17.55 \\ 0 & 1 & 0 & -\frac{9}{2} & -3.65 \\ 0 & 0 & 1 & -\frac{5}{2} & 2.3 \\ 0 & 0 & 0 & -\frac{15}{2} & 4.6 \end{array}\right]

STEP 10

Next, ensure the leading coefficient (pivot) in the fourth row is 1. Currently, it is −152-\frac{15}{2}, so divide the fourth row by −152-\frac{15}{2}.
R4=−215R4 R_4 = -\frac{2}{15} R_4
[100132−17.55010−92−3.65001−522.30001−4615] \left[\begin{array}{cccc|c} 1 & 0 & 0 & \frac{13}{2} & -17.55 \\ 0 & 1 & 0 & -\frac{9}{2} & -3.65 \\ 0 & 0 & 1 & -\frac{5}{2} & 2.3 \\ 0 & 0 & 0 & 1 & -\frac{46}{15} \end{array}\right]

STEP 11

Use row operations to make all other entries in the fourth column zero. - Subtract 132\frac{13}{2} times the fourth row from the first row. - Add 92\frac{9}{2} times the fourth row to the second row. - Add 52\frac{5}{2} times the fourth row to the third row.
R1=R1−132R4R2=R2+92R4R3=R3+52R4 R_1 = R_1 - \frac{13}{2} R_4 \\ R_2 = R_2 + \frac{9}{2} R_4 \\ R_3 = R_3 + \frac{5}{2} R_4
[1000−101000001000001−4615] \left[\begin{array}{cccc|c} 1 & 0 & 0 & 0 & -1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & -\frac{46}{15} \end{array}\right]

STEP 12

Read off the solutions from the RREF matrix.
The solutions are: x1=−1x2=0x3=0x4=−4615 x_1 = -1 \\ x_2 = 0 \\ x_3 = 0 \\ x_4 = -\frac{46}{15}

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