Applications of Linear Systems: Constraints and Viable Solutions
Standard
A.CED.3 Represent constraints by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.
Why is this important?
Sometimes when we come up with these systems of equations, negative numbers might not be viable solutions. For example, we might not want to consider negative time.
What is a viable solution?
A viable solution is a solution that makes sense within the context of the question.
What do we mean by constraints?
Constrain means to restrict or restrain. We want to restrict our range and domain to a certain interval that makes sense.
Example 1
pg 257
The zoo has two water tanks that are leaking. One tank contains 12 gal of water and is leaking at a constant rate of 3 gal/h. The second tank contains 8 gal of water and is leaking at a constant rate of 4 gal/h. When will the tanks have the same amount of water? Explain.
Create our equations
Tank 1
Starts at 12 gallons
Loses at a rate of 3 gallons per hour
Let y = gallons
Let x = hours
Equation:
- y = 12 - 3x
Tank 2
Starts at 8 gallons
Loses at a rate of 4 gallons per hour
Let y = gallons
Let x = hours
Equation:
- y = 8 - 4x
Graphing would be the best option to solve for this problem as you will want to see it visually.
Do we have a viable solution?
Look at the intersection of the two lines. The point is (-4, 24). x = -4. We do not consider negative time. y = 24. Neither tank ever had or will ever have 24 gallons of water. This is not a viable solution.
What are the constraints?
So we want to restrict or constrain ourselves from negative numbers. So we're looking at x = 0 and above. For y, we're considering y between 0 and 12. We cannot have a negative amount of water.
Our solution:
As you can see from Figure B, the lines do not meet. So the tanks never have the same amount of water until they are both empty (which can be thought of as having 0 amount of water).
Look at the blue graph which represents Tank 2. Click on that point where the blue graph meets the x axis, which is when y = 0. That coordinate is (2, 0), which means Tank 2 is empty after 2 hours.
Tank 1's coordinate at y = 0 is (4, 0), meaning that Tank 1 is empty after 4 hours.
So therefore, the tanks are both empty once Tank 1 is empty at 4 hours.