7x-3y=41
2x=-5y
Solve using augmented matrix
What do i do?
+7 -3 41
+2 +5 00
http://www.gregthatcher.com/Mathematics/GaussJordan.aspx
You should get;
1 0 5
0 1 -2
To solve the given system of equations using an augmented matrix, you need to perform row operations to transform the augmented matrix into its reduced row-echelon form (also known as the row-reduced echelon form or RREF). Here's how to do it step by step:
Step 1: Write down the augmented matrix of the given system of equations:
| 7 -3 | 41 |
| 2 -5 | 0 |
Step 2: Perform row operations to establish zeros below the pivot of the first column (the leftmost non-zero entry). The goal is to make the first entry in the second row (2, -5) become zero.
- Multiply the first row by 2 and the second row by 7.
- Subtract the second row from the first row.
The new augmented matrix after Step 2 is:
| 18 1 | 41 |
| 0 -39 | -82 |
Step 3: Perform row operations to establish a leading coefficient of 1 in the second row. We divide the second row by -39 to achieve this.
- Divide the second row by -39.
The new augmented matrix after Step 3 is:
| 18 1 | 41 |
| 0 1 | 2 |
Step 4: Perform row operations to establish zeros above and below the pivot of the second column. The goal is to make the entry in the first row, second column (1, 1) become zero.
- Multiply the second row by -1 and add it to the first row.
The new augmented matrix after Step 4 is:
| 18 0 | 43 |
| 0 1 | 2 |
Step 5: Perform row operations to establish a leading coefficient of 1 in the first row. We divide the first row by 18 to achieve this.
- Divide the first row by 18.
The new augmented matrix after Step 5 is:
| 1 0 | 43/18 |
| 0 1 | 2 |
The augmented matrix is now in reduced row-echelon form. The first column corresponds to the variable x, and the second column corresponds to the variable y. The rightmost column represents the constant term.
The solution to the system of equations is:
x = 43/18
y = 2
Thus, the solution to the system of equations, as obtained using the augmented matrix method, is x = 43/18 and y = 2.