Final answer:
To find the general solution using the elimination method, we start by manipulating the equations to eliminate a variable. Then, we solve the resulting equation to get the general solution for y. The general solution is given by the equation y = (2mx + 2c - 48x) / 3.
Step-by-step explanation:
To solve the given linear system using the elimination method, we need to eliminate one variable by manipulating the equations. Let's start by multiplying the first equation by 2: 2y' - 4y = 32x. Now, add this equation to the second equation (y' + 2y = 16x) to eliminate the 'y' term: 2y' - 4y + y' + 2y = 32x + 16x. Simplifying the left side gives us 3y' - 2y = 48x, which is the equation we can solve for the general solution.
Example: If we assume y' = k and y = mx + c, we can substitute these values into the equation to find the general solution.
3k - 2(mx + c) = 48x. Rearranging the equation, we get: 3k - 2mx - 2c = 48x. Now, we can separate the terms with 'k' and 'x' to get the general solution for y: y = (2mx + 2c - 48x) / 3.