3x-3y=12
3x+2y=-3
2x+y=4
Solve each by reduced row echelon form and if no solution, say it is inconsistent.
To solve a system of linear equations using reduced row echelon form, we use a process called Gaussian elimination. The goal is to transform the augmented matrix of the system into its reduced row echelon form. Here's how we can solve each of the given equations:
Given system of equations:
3x - 3y = 12 ... (1)
3x + 2y = -3 ... (2)
2x + y = 4 ... (3)
Step 1: Write down the augmented matrix of the system.
[3 -3 | 12]
[3 2 | -3]
[2 1 | 4 ]
Step 2: Apply row operations to transform the matrix into its reduced row echelon form.
Start by making the first element in the first row equal to 1 by dividing the first row by 3:
[1 -1 | 4 ]
[3 2 | -3]
[2 1 | 4 ]
Multiply the first row by 3 and subtract it from the second row to eliminate the x variable:
[1 -1 | 4 ]
[0 5 | -15]
[2 1 | 4 ]
Multiply the first row by 2 and subtract it from the third row to eliminate the x variable:
[1 -1 | 4 ]
[0 5 | -15]
[0 3 | -4 ]
Divide the second row by 5 to make the leading coefficient equal to 1:
[1 -1 | 4 ]
[0 1 | -3 ]
[0 3 | -4 ]
Multiply the second row by 3 and subtract it from the third row to eliminate the y variable:
[1 -1 | 4 ]
[0 1 | -3 ]
[0 0 | 5 ]
Step 3: Interpret the row echelon form to find the solutions.
From the row echelon form, we can see that the last row [0 0 | 5] implies 0 = 5, which is a contradiction. This contradicts the assumption that the system has a solution, indicating that the system is inconsistent. Therefore, there are no solutions to the given system of equations.
In conclusion, the given system of equations is inconsistent, meaning no solution exists.