write and row reduce the augmented matrix to find the general solution:
x - 2y + 13 = 0
y - 4x = 17
(1 -2 -13)
(-4 1 17)
(4 -8 -52)
(-4 1 17)
add to get
(4 -8 52)
(0 -7 -35)
divide to get 1's on the diagonal
(1 -2 -13)
(0 1 5)
add twice row 2 to row 1 to get
(1 0 -3)
(0 1 5)
To find the general solution of the given system of equations, we can write them in the form of an augmented matrix and then row reduce the matrix using the Gauss-Jordan elimination method.
First, let's rewrite the system of equations in matrix form:
1 -2 13 | 0
-4 1 0 | 17
Now we can start row reducing the augmented matrix. The goal is to transform the matrix into its reduced row echelon form.
1. Perform row operations to create zeros below the first entry of the first column by adding multiples of row 1 to other rows:
1 -2 13 | 0
0 -7 52 | 17
2. Multiply row 2 by (-1/7) to make the second entry of the second row equal to 1:
1 -2 13 | 0
0 1 -52/7 | -17/7
3. Perform row operations to create a zero above the second entry of the second column by adding 2 times row 2 to row 1:
1 0 117/7 | 34/7
0 1 -52/7 | -17/7
The augmented matrix is now in its row reduced echelon form. Let's convert it back into equation form:
x = 117/7 - 34/7y
y = -52/7 + 17/7y
To find the general solution, we can express one variable in terms of the other. Let's solve the second equation for y:
y = -52/7 + 17/7y
Multiply both sides of the equation by 7 to eliminate the denominators:
7y = -52 + 17y
Rearrange the terms:
17y - 7y = -52
Combine like terms:
10y = -52
Divide both sides of the equation by 10:
y = -5.2
Now substitute the value of y back into either of the original equations to solve for x:
x - 2(-5.2) + 13 = 0
x + 10.4 + 13 = 0
x + 10.4 + 13 = 0
x + 23.4 = 0
x = -23.4
Therefore, the general solution to the system of equations is:
x = -23.4
y = -5.2