write the system of equations in row-echelon form
2x+4y+z=-4
2x-4y+6z=13
4x-2y+z=6
To convert a system of equations into row-echelon form, we will perform row operations until the matrix representing the system satisfies the following conditions:
1. The first non-zero entry in each row (called a leading entry) is to the right of the leading entry in the row above it.
2. All rows consisting of only zeros are at the bottom, if any.
3. If a row contains only zeros, all rows below it must also consist only of zeros.
Here's how we can transform the given system of equations into row-echelon form:
Step 1: Write the augmented matrix for the system.
The augmented matrix for the given system is:
[ 2 4 1 | -4 ]
[ 2 -4 6 | 13 ]
[ 4 -2 1 | 6 ]
Step 2: Perform row operations to obtain zeros below the first leading entry in each row.
R2 = R2 - R1 (subtract the first row from the second row)
R3 = R3 - 2R1 (subtract two times the first row from the third row)
The matrix becomes:
[ 2 4 1 | -4 ]
[ 0 -8 5 | 17 ]
[ 0 -10 -1 | 14 ]
Step 3: Normalize the second row by dividing it by its leading coefficient.
R2 = (-1/8)R2 (divide the second row by -8)
The matrix becomes:
[ 2 4 1 | -4 ]
[ 0 1 -5/8| -17/8]
[ 0 -10 -1 | 14 ]
Step 4: Perform row operations to obtain zeros below the second leading entry in each row.
R3 = R3 + 10R2 (add ten times the second row to the third row)
The matrix becomes:
[ 2 4 1 | -4 ]
[ 0 1 -5/8| -17/8]
[ 0 0 -21/8| -16/8 ]
We have obtained the row-echelon form of the given system of equations. The matrix representation of the row-echelon form is:
[ 2 4 1 | -4 ]
[ 0 1 -5/8| -17/8]
[ 0 0 -21/8| -16/8 ]
Now, we can write the corresponding system of equations:
2x + 4y + z = -4
y - (5/8)z = -(17/8)
-(21/8)z = -(16/8)
We can simplify the last equation to -21z = -16. Dividing by -21 gives us z = 16/21.
Substituting the value of z into the second equation, we get y - (5/8)(16/21) = -(17/8). Simplifying further, we have y - 40/168 = -(17/8). To make the calculation easier, we can reduce 40/168 to 5/21. Therefore, y - 5/21 = -(17/8). To get rid of fractions, we can multiply the entire equation by 168, resulting in 168y - 40 = -(357/8).
Similarly, substituting the value of z into the first equation, we get 2x + 4y + (16/21) = -4. Rearranging terms, we have 2x + 4y = -4 - (16/21). Simplifying further, we can reduce -4 - (16/21) to -88/21. The equation becomes 2x + 4y = -(88/21).
So, the system of equations in row-echelon form is:
2x + 4y = -(88/21)
y - (5/8)z = -(17/8)
z = 16/21