Final answer:
To find the corner points of the system of inequalities, we turn the inequalities into equations, solve them in pairs to find intersection points, and then verify these points against the original inequalities to confirm they lie within the feasible region.
Step-by-step explanation:
To find the coordinates of each corner point in the given system of inequalities, we need to determine where the lines created by turning each inequality into an equation intersect. The inequalities given are:
- 16x + 13y ≤ 91,
- 3x + 4y ≥ 18,
- -4x + 3y ≤ 12.
To find the intersection points, we first turn these inequalities into equations:
- 16x + 13y = 91,
- 3x + 4y = 18,
- -4x + 3y = 12.
Then, we solve these equations in pairs to get the corner points of the feasible region. The solutions of these equations give us the corner points.
For example, to solve the first two equations, we can use substitution or elimination methods. Suppose we multiply the second equation by 4 to get 12x + 16y = 72 and then subtract it from the first equation multiplied by 3, yielding:
- 48x + 39y = 273
- -(12x + 16y = 72)
Subtracting these equations gives us:
From which we can solve for y and then for x.
Repeat this process for the other pairs of equations and check if the corner points satisfy all the original inequalities to ensure they belong to the feasible region.