Solving a System of Equations - Math Problem Example

Final Project The system of linear equations may be applied to mixture problems in which itemsof various types or grades are sold at distinct values. This is particularly useful for starting entrepreneurs who consider establishing business on a small-scale with simple calculations carried out. Application: Matthew would like to put up a small coffee-bean store selling two special varieties of coffee – Arabica(M) and Robusta(H). He estimates obtaining profit from each kg of Arabica(M) worth $10.50 and from each kg of Robusta(H) costing $9.25. If he desires to have 2500 kgs of bean-mixture sold for $9.74 per kg, how much of each kind must be present in the mixture? Solution: Let x = the quantity of Arabica(M) (in kg) and y = quantity of Robusta(H) (in kg) Based on the given information, equations may be set up as: 10.50(x) + 9.25(y) = (9.74)(2500) ---? equation (1) x + y = 2500 ---? equation (2) Graphing each equation on the same xy-plane: By applying substitution method (equation (2) into equation (1)): 10.50(x) + 9.25*(2500 – x) = (9.74)(2500) 10.50(x) + 23125 - 9.25(x) = 24350 1.25(x) = 1225 Then dividing each side by 0.8, x = 980 kgs Arabica(M) And 980 + y = 2500 ---? y = 1520 kgs Robusta(H) Thus, the point of intersection is at (980, 1520) and this pertains to the quantities each of the Arabica(M) and the Robusta(H) that must be present in the bean-mixture so that Matthew is able to satisfy the condition of selling a total of 2500-kg mixture where each kilogram is sold for $9.74. Summary of Learning Besides its flexible range of applications, I have learned that there can be alternative methods in solving a system of equations once each equation has been properly set up with correct algebraic expressions in which variables are made to represent unknown amounts of objects either count or non-count by nature. I appreciate the fact that in Algebra, one is able to verify the existence of a solution by using methods of elimination and substitution wherein one method can be a means to countercheck the other which ought to show the same results. It is quite interesting that equations may be graphed to determine whether real solutions exist as via intersection of lines. Having become acquainted with different function types such as linear, polynomial, rational, logarithmic, and exponential, I gain knowledge of constructing relations among dependent and independent variables as well as arbitrary constants based on useful empirical data. Summary of Topic In the model mixture problem, businessmen like Matthew can set constraints in terms of cost, quantity of material or commodity under consideration, selling price, and additional concerns that may possibly be incorporated in formulating labels and pertinent equations. Normally, problems of such kind possess linear relationships of variables for which the number of solutions rely on the highest degree of independent variable by which to identify the number of intersection points between the set of equations involved. Alternative Project In its existence and approach, Algebra serves as a base device to higher math such as Calculus which attempts to explore the grounds for the undefined nature of a function and designates a sensible understanding about up to which extent it would exist considering assumptions or applicable conditions. Fundamentals of algebra are essential to the foundation of courses designed to solve multivariable systems through linear programming, matrix applications, and differential equations where there is ceaseless necessity for equations and functions in interpreting problem situations. They are especially of ample advantage as tools for working chemists and biochemists who deal with cases of radioactive decomposition or rates of reactions for instance. Hence, chemical studies under such field may include the use of exponential function A = A0*e-kt where ‘A’ stands for the element concentration or amount at any time ‘t’ whereas ‘A0’ refers to the initial amount with ‘k’ as the constant rate. This may be assumed if the experimenter finds that the actual data obtained coincide with a graphical trend that is close to an exponential behavior as such. Culturing of bacteria and isotope dating of a certain element are two areas that have sought wide interest in working with exponentially and logarithmically conveyed mathematical relationships. The same is true for a chemical engineer who works to design a reactor by optimizing dimensions according to reaction mechanisms as predicted by the order of reaction rate which is often encountered in a linear setup of the form ln [A] = -kt + ln [A0] whereby this time ‘A’ makes reference to the molarity of a disappearing reactant with time in a typically catalyzed setting. Though computations are naturally theoretical, they supply a rough estimate of values that can more or less be expected to match laboratory outcomes. Principles of algebra also allow students to learn about concepts on domain and range and how a function may come really close or approach a value at least even if it is never meant to cross its exact location when sketched. Scientists would not have discovered azeotropes and their essence in distillation without acquiring knowledge on asymptotes firsthand since these azeotropes indicate the feasibility or the extent to which separation of components may proceed. As a concrete instance, in ethanol-water mixture, it is found that the pinch of azeotrope is at approximately 87% of ethanol which means that this is the highest achievable percent purity of ethanol as derived from its mixture with water due to close boiling points of the two substances, unless a third component is incorporated to break the asymptotic or limiting property. Algebra in plain everyday living, on the other hand, may be found being utilized in circumstances when some employees of a specific workplace compare each other’s capacity in accomplishing a common task. For instance, if worker A is capable of completing a job of meeting a quota out of doing item collections for x-hours while it takes worker B to finish the same job in y-hours then, the two colleagues may play with the curiosity of figuring out the z-hours it would take if they occur to work together at equal rates which can be related as – (1/x) + (1/y) = (1/z) Of course, the basic assumption of uniform rate for both must be made initially in order to justify the validity of the mathematical statement generated. That manner, either employee can impact time management with efficiency on keeping track of and adjusting hours of work on the basis of individual load per unit time. Then another practical aspect which operates similarly with the rate of work equation is in measuring pipe flows that may or may not account for algebraically manipulated factors like leaks, enabling civil engine specialists, for example, to yield assistance for their structural analysis. Read More