Final answer:
To solve the given system of equations, we can use the method of elimination. Multiply the equations by appropriate constants to eliminate one variable, combine the equations, and solve for the remaining variable. The solutions are x < 6/5 and y > 2/5.
Step-by-step explanation:
To find the solutions to the given system of equations, we need to solve them simultaneously. The given equations are:
2x+3y > 6
3x+2y ≤ 6
To solve them, we can use the method of substitution or elimination. Let's use the method of elimination to eliminate one variable and solve for the other.
- Multiply the first equation by 2 and the second equation by 3 to make the coefficient of 'y' equal in both equations. This will help us eliminate 'y' when we add the equations together.
- The modified equations become: 4x + 6y > 12 and 9x + 6y ≤ 18.
- Now, subtract the second equation from the first equation: (4x + 6y) - (9x + 6y) > 12 - 18.
- Simplifying the equation gives -5x > -6.
- Divide both sides of the equation by -5 to solve for 'x'. The inequality becomes x < 6/5.
- Now, substitute the value of 'x' into one of the original equations to solve for 'y'. Let's substitute x = 6/5 into the first equation: 2(6/5) + 3y > 6.
- Simplifying the equation gives 12/5 + 3y > 6.
- Subtract 12/5 from both sides of the equation: 3y > 6 - 12/5.
- Simplifying the equation gives 3y > 18/5 - 12/5.
- Combine the terms on the right side of the equation: 3y > 6/5.
- Divide both sides of the equation by 3 to solve for 'y'. The inequality becomes y > 2/5.
Therefore, the solutions to the given system of equations are x < 6/5 and y > 2/5.