How do I use gauss-jordan to solve this? Can you explain the steps and show what you multiplied, added, subtracted, etc to each row?
8x + 2y = -68
-7x - y = 55
augmented matrix
+8 +2 -68
-7 -1 +55
you have two rows, three columns
now go to
http://www.gregthatcher.com/Mathematics/GaussJordan.aspx
-7
-6
divide first row by 8 to get 1 in upper left
1 1/4 -17/2
add 7 times new first row to second to get 0 below the one
1 +1/4 -17/2
0 +3/4 -9/2
subtract 1/3 times row 2 from row 1 to get rid of that 1/4 in row 1
1 0 -7
0 3/4 -9/2
multiply second row by 4/3 to get 1 in 2,2 spot
1 0 -7
0 1 -6
done
1 x + 0 y = -7
0 x + 1 y = -6
To solve the given system of equations using Gauss-Jordan elimination, we need to perform a series of row operations until we reach row-echelon form or reduced row-echelon form. Here are the step-by-step calculations:
Step 1: Write the system of equations in augmented matrix form:
[ 8 2 | -68 ]
[ -7 -1 | 55 ]
Step 2: Perform row operations to transform the matrix into row-echelon form.
Multiply the first row by -7 and add it to the second row:
[ 8 2 | -68 ]
[ 0 15 | -429 ]
Now, divide the second row by 15 to simplify:
[ 8 2 | -68 ]
[ 0 1 | -429/15 ]
Step 3: Perform additional row operations to achieve reduced row-echelon form.
Multiply the second row by 2 and subtract it from the first row:
[ 8 0 | -68 + 2*(-429/15) ]
[ 0 1 | -429/15 ]
Simplify the first row:
[ 8 0 | -68 + (-286/15) ]
[ 0 1 | -429/15 ]
Further simplification gives:
[ 8 0 | -1398/15 ]
[ 0 1 | -429/15 ]
Step 4: Now, the matrix is in reduced row-echelon form, and we can read off the solutions from it.
From the first row, we obtain the value for x:
8x = -1398/15
x = -1398/15 * 1/8
x = -232.5
From the second row, we obtain the value for y:
y = -429/15
So, the solution to the system of equations is x = -232.5 and y = -429/15.
During the process, we multiplied the rows by specific values and subtracted multiples of one row from another to simplify the system and eventually obtain the values for x and y.