Assignment
Part I: A system of equations can result in 3 possibilities: consistent, inconsistent, and dependent. Discuss each of these types of solution that can be generated, and describe the graph of each possibility.
Part II: Break-Even Analysis and Cost Analysis: As a first-time homeowner, there are going to many decisions that you need to make, such as whom to hire for the upkeep of your lawn. You have just received advertisements from 2 companies in your area: Greener Lawns and Lawns for Less. Greener Lawns charges an initial fee of $200, plus $50 per hour for labor costs. Lawns for Less charges an initial fee of $300, plus $25 per hour for labor costs.
1.If h represents the number of labor hours and C represents the cost, write the total cost equation for Greener Lawns.
2.If h represents the number of labor hours and C represents the cost, write the total cost equation for Lawns for Less.
3.Solve the system of equations for the total cost of lawn care using the desired technique: substitution, elimination, or graphing.
4.Document how you came to these conclusions for accuracy.
Mathematics of Finance
Part III: To purchase your first home, you may be required to borrow funds from a bank. You have just graduated from college, and your dream is to own your first home. Before you begin looking for your dream home, you need to learn more about funding options and the process required to finance a home.
Research online to find more information about home loans and mortgages.
1.Discuss your options for obtaining a home loan and how mortgages work.
2.Discuss the process/procedures for obtaining the loan and the ideal interest rates for home loans.
3.Report your findings in 2 paragraphs.
Part I:
When solving a system of equations, there are three possibilities for the solutions: consistent, inconsistent, and dependent.
1. Consistent solution: A system of equations has a consistent solution if it has a unique solution, meaning that there is a set of values for the variables that satisfies all of the equations. Geometrically, this is represented by the intersection point(s) of the graphs of the equations. Each equation represents a line on a graph, and the consistent solution is the point where these lines intersect. In this case, the system can be solved by any of the three methods: substitution, elimination, or graphing.
2. Inconsistent solution: A system of equations has an inconsistent solution if there is no set of values for the variables that satisfies all of the equations. Geometrically, this is represented by parallel lines on the graph. If the lines never intersect, it means that the system of equations has no solution. In this case, the system is inconsistent, and there is no way to solve it.
3. Dependent solution: A system of equations has a dependent solution if there are infinitely many solutions. Geometrically, this is represented by overlapping lines on the graph. When the equations represent the same line, they have the same solution for every possible value of the variables. In this case, the system is dependent, and the equations are essentially expressing the same relationship in different forms. The two equations are not independent and can be combined into a single equation. Graphically, the lines representing the equations will completely overlap each other.
Part II:
1. The total cost equation for Greener Lawns can be written as C = 200 + 50h, where C represents the total cost and h represents the number of labor hours. The initial fee of $200 is added to the product of $50 and the number of labor hours to get the total cost.
2. The total cost equation for Lawns for Less can be written as C = 300 + 25h, where C represents the total cost and h represents the number of labor hours. The initial fee of $300 is added to the product of $25 and the number of labor hours to get the total cost.
3. To solve the system of equations for the total cost of lawn care, we can use the desired technique: substitution, elimination, or graphing. We can substitute one equation into the other and solve for one variable, then substitute that value back into the other equation to find the other variable. Alternatively, we can eliminate one variable by manipulating the equations and then solve for the remaining variable. Graphing involves graphing both equations and finding the point of intersection.
4. To accurately solve the system of equations, you can take the total cost equations for Greener Lawns and Lawns for Less, set them equal to each other, and solve for the number of labor hours h. Once you have the value for h, you can substitute it back into either equation to find the corresponding total cost C.