rst Order DEs Ch 2 Higher Order DEs Ch 3 Laplace Transform L

rst Order DEs Ch 2: Higher Order DEs Ch 3: Laplace Transform Linear Systems 4. Solve the system x 5y 3x -y where x (0) F 0 and y(0) F 2 by (a) The Elimination Method (b) The Eigenvector/Eigenvalue Method (c) The Laplace Transform Method

Solution

A way to solve a linear system algebraically is to use the substitution method. The substitution method functions by substituting the one y-value with the other. We\'re going to explain this by using an example.

y=2x+4y=2x+4

3x+y=93x+y=9

We can substitute y in the second equation with the first equation since y = y.

3x+y=93x+y=9

3x+(2x+4)=93x+(2x+4)=9

5x+4=95x+4=9

5x=55x=5

x=1x=1

This value of x can then be used to find y by substituting 1 with x e.g. in the first equation

y=2x+4y=2x+4

y=21+4y=21+4

y=6y=6

The solution of the linear system is (1, 6).

You can use the substitution method even if both equations of the linear system are in standard form. Just begin by solving one of the equations for one of its variables.