Write the system of equations as an augmented matrix. Then solve for x and y.
3x+y=-11
-9x-3y=33
No unique solution, equations are the same
+3 +1 -11
-9 -3 +33
http://www.idomaths.com/gauss_jordan.php
1 0 -3.667
0 0 0
x = -3.6666.... or -11/3
y can be anything
equation 2 is -3 times equation 1
Therefore the two lines are the same, they do not intersect, there is no solution
To write the system of equations as an augmented matrix, we can arrange the coefficients of x and y as the entries of a matrix, and the constant terms on the right side of the equations as a separate column. The matrix is augmented by drawing a vertical line to separate the coefficient matrix from the constant column.
The given system of equations is:
3x + y = -11 ...(1)
-9x - 3y = 33 ...(2)
The corresponding augmented matrix can be written as:
[ 3 1 | -11 ]
[-9 -3 | 33 ]
To solve for x and y using this augmented matrix, we will perform row operations to transform the matrix into a reduced row-echelon form. Here are the steps:
Step 1: Multiply row 1 by 3.
[ 3 1 | -11 ]
[-9 -3 | 33 ]
--------------------------------
[ 9 3 | -33 ]
[-9 -3 | 33 ]
Step 2: Add row 1 to row 2.
[ 9 3 | -33 ]
[-9 -3 | 33 ]
--------------------------------
[ 9 3 | -33 ]
[ 0 0 | 0 ]
Step 3: Divide row 1 by 3.
[ 9/3 3/3 | -33/3 ]
[ 0 0 | 0 ]
--------------------------------
[ 3 1 | -11 ]
[ 0 0 | 0 ]
The reduced row-echelon form of the augmented matrix shows that the system of equations is dependent, meaning there are infinite solutions.
To solve for x and y, we will express y in terms of x. From the reduced row-echelon form, we can see that x is a leading variable (since it corresponds to a pivot column) and y is a free variable (as it corresponds to a non-pivot column). We can write the solution as:
x = t (a parameter or constant)
y = -3t - 11 (using the equation from row 1)
So, there are infinite solutions given by the parametric equations x = t, y = -3t - 11, where t can take any real value.