Nicole decides to make two sizes of bottles of BBQ sauce, 16 ounces and 8 ounces. Her total production is 1,280 ounces for the summer. She has room for up to 100 bottles in her booth at the farmers market and wants to maximize her income from their sale. Let x represent the number of 16-oz bottles sold and y represent the number of 8-oz bottles. The 16-oz bottles sell for $25 and the 8-oz bottles sell for $15.

Step-by-step explanation:

Given that Nicole decides to make two sizes of bottles of BBQ sauce, 16 ounces and 8 ounces. Her total production is 1,280 ounces for the summer. She has room for up to 100 bottles in her booth at the farmers market and wants to maximize her income from their sale

Let x be the 16 oz bottle and y the 8 oz bottle

Profit Z = 25x+15y and objective is to maximize Z

constrains are

Solving we get point of intersectin is x = 60 and y =40

Other corner points are x = 80, y=0 or x=0, y =100

Profit for (60,40) is 25(60)+15(40) = 2100

Profit for (80,0) is 25(80) = 2000

Profit for (0,100) is 1500

So maximum profit at 60, 40 for 16 oz and 8 oz

(0,-3) see attached graph

step-by-step explanation:

a. to solve the system, graph the two equations.

x = y+3 becomes y=x-3 has m=1/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move up 1 unit and 1 unit to the right. plot a point and connect.

y=-4x+3 has m =-4/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move down four units and to the right 1 unit. plot a point and connect.

the solution is the point (x,y) where the two lines intersect. by the graph below it is (0,-3).

b. to solve by substitution, substitute y = -4x-3 into x= y+3

x= (-4x-3) +3

x=-4x

5x=0

x=0

now substitute x = 0 back in to find y.

y= -4(0)-3

y=0-3

y=-3

solution is (0,-3)

c. use elimination by adding the equations together.

x-y=3

4x+y=-3

5x=0

x=0

then substitute into one equation to find y.

y=-4(0)-3

y=-3