Graph the system of constraints and find the value of x and y that maximize the objective function.

Question

constraints { x>=0 }
{ y>= 0 }
{ y<=3 }
{y<= -x + 5 }
Objective function on: C=-5x+3y

Answer 1

To graph the system of constraints, we start by graphing each individual constraint on the xy-plane.

1. x >= 0:
This constraint indicates that x must be greater than or equal to 0, which means all points to the right of the y-axis are valid solutions. We shade the area to the right of the y-axis.

2. y >= 0:
This constraint indicates that y must be greater than or equal to 0, which means all points above the x-axis are valid solutions. We shade the area above the x-axis.

3. y <= 3:
This constraint indicates that y must be less than or equal to 3, which means all points below or on the line y = 3 are valid solutions. We draw a solid line at y = 3 and shade the area below it.

4. y <= -x + 5:
This constraint indicates that y must be less than or equal to -x + 5, which means all points below or on the line y = -x + 5 are valid solutions. We draw a solid line with a slope of -1 passing through the point (5, 0) and shade the area below it.

Now, we identify the feasible region, which is the shaded area that satisfies all the constraints. In this case, it is the triangle formed by the intersection of the shaded areas.

To find the value of x and y that maximize the objective function, we need to find the vertex of the feasible region that maximizes the objective function C = -5x + 3y. Since the objective function is linear, it will have the maximum value at one of the extreme points of the feasible region.

Upon analyzing the graph, we can see that the vertex located at (0,3) is the highest point within the feasible region. Plugging in these values into the objective function, we find:

C = -5(0) + 3(3) = 9

Therefore, the maximum value of the objective function is C = 9, and the values of x and y that maximize it are x = 0 and y = 3.

Answer 2

How can you represent the system of equations with a matrix?

{11x-12y=7
{5x+12y=-7

Answer 3

To represent the system of equations with a matrix, we can write the coefficient matrix and the constant matrix.

The coefficient matrix is created by taking the coefficients of x and y terms and arranging them in a matrix format. In this case, the coefficient matrix is:

[[11, -12],
[5, 12]]

The constant matrix is created by taking the constants on the right side of the equations and arranging them in a matrix format. In this case, the constant matrix is:

[[7],
[-7]]

Now, let's represent the system of equations using matrices:

Coefficient matrix * Variable matrix = Constant matrix

[[11, -12],
[5, 12]] * [[x],
[y]] = [[7],
[-7]]

When you multiply the coefficient matrix by the variable matrix, you obtain the constant matrix, representing the system of equations.

Answer 4

Use a matrix to find the solution to the system of equations.

{-8x-8y=-16
{6x-9y=-108

Answer 5

To find the solution to the system of equations using a matrix, we will represent it in matrix form.

We start by creating the coefficient matrix, variable matrix, and constant matrix:

Coefficient matrix:
[[ -8, -8],
[ 6, -9]]

Variable matrix:
[[ x],
[ y]]

Constant matrix:
[[ -16],
[-108]]

We can rewrite the system of equations in matrix form as:

Coefficient matrix * Variable matrix = Constant matrix

[[ -8, -8],
[ 6, -9]] * [[ x],
[ y]] = [[ -16],
[-108]]

To solve for the variable matrix, we can multiply both sides of the equation by the inverse of the coefficient matrix. The inverse of a 2x2 matrix can be found using the following formula:

A^(-1) = (1/det(A)) * [[ d, -b],
[-c, a]]

where A is the 2x2 matrix, det(A) is the determinant of A, and a, b, c, and d represent the elements of A.

For our coefficient matrix, the determinant can be calculated as (-8)(-9) - (-8)(6) = -72 + 48 = -24.

So, the inverse of the coefficient matrix is:

(1/-24) * [[ -9, 8],
[ -6, -8]]

Multiplying both sides of the equation by the inverse of the coefficient matrix, we have:

(1/-24) * [[ -9, 8],
[ -6, -8]] * [[ -8, -8],
[ 6, -9]] * [[ x],
[ y]] = (1/-24) * [[ -9, 8],
[ -6, -8]] * [[ -16],
[-108]]

Simplifying the equation, we have:

[[ x],
[ y]] = [[ 2],
[ 2]]

Therefore, the solution to the system of equations is x = 2 and y = 2.