Material cost equation : 30x + 2y = 200
Profit equation : 100x + 5y = 400
B. Explain your inequalities and explain why x<8 and y<40 are also inequalities for this system
In this system of equations, we have two equations representing the material cost and the profit. The equations are as follows:
Material cost equation: 30x + 2y = 200
Profit equation: 100x + 5y = 400
To understand why x < 8 and y < 40 are inequalities for this system, let's analyze the equations further.
1. Material cost equation: 30x + 2y = 200
This equation represents the relationship between the quantities of two materials, x and y, and the total cost of these materials. The left side of the equation represents the cost, which is determined by multiplying the quantity of material x (30x) and the quantity of material y (2y). The right side of the equation represents the total cost, which is 200.
In this equation, the coefficients 30 and 2 represent the cost of each unit of x and y respectively. Since the cost of material y is only 2, compared to 30 for material x, it means that material y is cheaper. Therefore, the cost of the materials is primarily influenced by the quantity of material x. This implies that as the quantity of x increases, the total cost will increase more significantly.
2. Profit equation: 100x + 5y = 400
This equation represents the relationship between the quantities of materials x and y and the resulting profit. The left side of the equation represents the profit, while the right side represents the target profit, which is 400.
Similar to the material cost equation, the coefficients in this equation represent the profit generated by each unit of x and y. In this case, the profit for material y is 5, compared to 100 for material x. This implies that the profit is more heavily influenced by the quantity of material x.
Now, let's discuss why x < 8 and y < 40 are inequalities for this system.
1. x < 8:
Since the cost equation primarily depends on the quantity of material x, we can determine that x < 8 because if we set x = 8 or higher, the material costs would exceed the budgeted total cost of 200. Therefore, to ensure that the total cost is within the given budget, the quantity of material x must be less than 8.
2. y < 40:
Similarly, considering the profit equation, which is mostly influenced by the quantity of material x, we can reason that y < 40. This is because if we set y = 40 or higher, the profit generated by the material y would exceed the target profit of 400. So, to meet the profit goal, the quantity of material y must be less than 40.
In summary, x < 8 and y < 40 are both inequalities because they represent the maximum quantity of materials x and y that can be used while still staying within the given constraints of the total cost and the target profit.