Answer
-$\frac{1}{2}$; the original solution is (5, 1); the new solution is (-5, -1)
Work Step by Step
Rewrite the system of equations as an augmented matrix:
{2 -4 | 6
-1 5 | 0}
Multiply the first row vector by -$\frac{1}{2}$, and use $R_{2}$-$R_{1}$=$R_{2'}$ to convert the system to a triangular matrix
{-1 5 | 0} - {-1 2 | -3} = {0 3 | 3}
Our new augmented matrix is now:
{2 -4 | 6
0 3 | 3}
Now that our augmented matrix is triangular, we can convert it back to a system of equations, and solve using back-substitution:
2x-4y=6
3y=3
y=1
Substituting y=1 into equation 1:
2x-4(1)=6
2x-4=6
2x=10
x=5
The solution is (5, 1)
If the system of equations were changed to:
2x-4y=-6
-x+5y=0
we can solve again using Gaussian elimination. The values of our matrix will be the same, except for value (1,3), which instead of 6 will be -6, and value (2,3), which instead of 3 will be -3. Our new triangular system of equations will be:
2x-4y=-6
3y=-3
and we can once again solve using back-substitution.
3y=-3
y=-1
Substituting y=-1 into equation 1:
2x-4(-1)=-6
2x+4=-6
2x=-10
x=-5
The new solution is (-5, -1)
