18. Systems of Equations and Matrices

Introduction to Matrices

18. Systems of Equations and Matrices

Introduction to Matrices

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Multiple Choice

Solve the system of equations by using row operations to write a matrix in REDUCED row-echelon form.
$4x+2y+3z=6$
$x+y+z=3$ $5x+y+2z=5$

$x=0,y=-3,z=4$

$x=-\frac14,y=-\frac{19}{4},z=\frac{11}{2}$

$x=1,y=4,z=-2$

$x=\frac12,y=\frac12,z=1$

Verified step by step guidance

Step 1: Write the system of equations as an augmented matrix. The system is: 4x + 2y + 3z = 6, x + y + z = 3, 5x + y + 2z = 5. The augmented matrix is: [[4, 2, 3, 6], [1, 1, 1, 3], [5, 1, 2, 5]].

Step 2: Use row operations to get a leading 1 in the first row, first column. You can do this by swapping the first row with the second row, resulting in: [[1, 1, 1, 3], [4, 2, 3, 6], [5, 1, 2, 5]].

Step 3: Eliminate the first column below the leading 1 by using row operations. Subtract 4 times the first row from the second row, and 5 times the first row from the third row. This gives: [[1, 1, 1, 3], [0, -2, -1, -6], [0, -4, -3, -10]].

Step 4: Get a leading 1 in the second row, second column by dividing the second row by -2. The matrix becomes: [[1, 1, 1, 3], [0, 1, 0.5, 3], [0, -4, -3, -10]].

Step 5: Eliminate the second column below the leading 1 in the second row by adding 4 times the second row to the third row. This results in: [[1, 1, 1, 3], [0, 1, 0.5, 3], [0, 0, -1, 2]]. Now, back substitute to find the values of x, y, and z.