use an augmented matrix to solve the system. what is the value of x and y ?
x+y=5
3x-y=-1
To solve the system of equations using an augmented matrix, we first convert the system into matrix form. The matrix form for this system would look like this:
```
[1 1 | 5]
[3 -1 |-1]
```
This matrix represents the coefficients of x and y in the first row and the second row, with the right side of the matrix representing the constants from the right-hand side of each equation.
Now, we'll use Gaussian elimination to convert this matrix into reduced row-echelon form.
Step 1: Make sure the pivot position of the first row (the position of the first "1") is set by having a non-zero number. In this case, we already have a "1" at the position for x in the first equation, so no change is necessary.
Step 2: Now we'll eliminate the x term from the second equation using row operations. We want to make the (3,1) element of the matrix (the "3" in the second row first column) turn into 0. To do this, we can replace the second row with the second row minus 3 times the first row:
```
Row_2 = [3 -1 |-1] - 3 * [1 1 | 5]
= [3 -1 |-1] - [3 3 | 15]
= [0 -4 |-16]
```
The updated augmented matrix is now:
```
[1 1 | 5]
[0 -4 |-16]
```
Step 3: We can now scale the second row to make the coefficient of y in the second equation be "1". Since the current coefficient is "-4", we can multiply the entire second row by -1/4:
```
Row_2 = (-1/4) * [0 -4 |-16]
= [0 1 | 4]
```
The augmented matrix updates to:
```
[1 1 | 5]
[0 1 | 4]
```
Step 4: Next, eliminate the y term from the first equation by replacing the first row with the first row minus the second row:
```
Row_1 = [1 1 | 5] - [0 1 | 4]
= [1 0 | 1]
```
The final augmented matrix is now:
```
[1 0 | 1]
[0 1 | 4]
```
This matrix is in reduced row-echelon form, and from here, we can read off the solutions directly:
x = 1
y = 4
Therefore, the solution to the system of equations is x = 1 and y = 4.