Intersection of Economic Function: Lines A Graphical Perspective on Elimination and Substitution Methods

Break Even Point in Business: Using the Substitution Method for Mathematical Analysis Name: Alvinska Hebat Grade: 910-Math Elimination and Substitution Methods: The Concept of Equilibrium and Break Even in Economics • • • • The elimination method and the substitution method are aimed at finding one value for each variable that satisfies two or more equations. The end result of both methods will have the same result, so we can choose one of them. Graphically, the elimination method and the substitution method will produce a single intersection point between the lines of the function equations we are using. The elimination method and the substitution method in economics are often used in the concept of equilibrium and break even, for example, to find the market equilibrium point and the break-even point (Break Even company). Elimination Method The elimination method is to eliminate one of the variables we choose by multiplying it by a certain number so that finally the variable is eliminated, and finally we can find out one value of the variables we use that satisfies certain equations. Elimination Method for One Free Variable When we have two equations as follows: Equation (1) : Y = f(X) → 5Y = 0.5X - 20 Equation (2): Y = g(X) →2Y = -0.8X + 12 If we choose to eliminate Y then equation (1) will be multiplied by 2 and equation (2) multiplied by 5 so that the results become as follows: Equation (1): 10Y = X - 40 Equation (2): 10Y = -4X + 60 - then Y can be found by 0 = 5X - 100 entering the value of X in 5X = 100 equation (1) or (2), eg: X = 100/5 = 20 5Y = 0.5X - 20 because X = 20 5Y = 0.5 (20) - 20 5Y = -10 → then Y = -10/5 = -2 From the results of this elimination calculation obtained (X, Y) = (20, -2) Elimination Method for Two Free Variables If we are going to do an elimination process where the equation we are using contains two independent variables then we need three equations to be eliminated. For example: Equation (1) : 2Y = 3X + 4Z - 40 Equation (2): 5Y = 7X - 2Z + 15 Equation (3): -6Y = 12X + 3Z - 20 In doing the elimination process we do it gradually, namely we do the elimination of equation (1) with equation (2) with the aim of eliminating Z, then the equation (1) multiplied by 1 and equation (2) multiplied by 2, becomes: Equation (1) : 2Y = 3X + 4Z - 40 Equation (2): 10Y = 14X - 4Z + 30 + 12Y = 17X - 10 ........... Equation(4) After that we eliminate Z from equations (1) and (3) or from equations (2) and (3). If we eliminate Z from equations (1) and (3) then we multiply equation ( 1) by 3 and equation (3) by 4, lOMoARcPSD|26370826 to be: Equation(1) → 6Y = 9X + 12Z - 120 Equation(3) → -24Y = 48X + 12Z - 80 - 30Y = -39X - 40 Pers(5) Then do the elimination of equations (4) and (5) by multiplying Equation(4) by 30 and multiplying equation (5) by 12 to become : Equation(4) → 360Y = 510X - 300 Equation(5) → 360Y = -468X - 480 - 0 = 978X + 180 → then X = -180/978 = -0.184 Enter price X = - 0.184 to equation (4) to find Y as follows: Equation (4) → 12Y = 17(- 0.184) - 10 12Y =13.1288 → then Y = -13.1288/12 = -1.094 To find the price of Z, enter the prices of X and Y into equation (1) or (2) or (3). If we put them into equation (1) then : Equation (1) → 2Y = 3X + 4Z - 40 2(-1.094) = 3(-0.184) + 4Z - 40 -2.188 = -0.552 + 4Z - 40 -4Z = -0.552 + 2.188 - 40 -4Z = 38.364 Z = 9.591 So (X,Z,Y) = (-0.184, 9,591, -1.094 ) Substitution Method The substitution method is used to find one value of each variable used through the process of replacing (substituting) the value of the variable we choose through other equations. Inorder to prove that the elimination result is the same as the substitution result, we use the example of the previous elimination problem. If we have two equations as follows: Equation (1): Y = f(X) → 5Y = 0.5X - 20 Equation (2): Y = g(X) →2Y = -0.8X + 12 Y = 0.1X - 4 We substitute Y = 0.1X - 4 to equation (2) so that equation (2) becomes as follows: 2 (0.1X - 4) = -0.8X + 12 0.2X - 8 = 0.8X + 12 0.2X + 0.8X = 12 + 8 then X = 20 Since X = 20 then Y can be found by putting the value of X into equation (1) or (2). If we enter the price of X into equation (1) then we get the following: 5Y = 0.5X - 20 5Y = 0.5 (20) - 20 5Y = -10 → means Y = -10/5 = -2 The result of this substitution calculation obtained (X,Y) = (20,-2) is the same as the elimination result. Application of Elimination and Substitution Method in Economic Concepts In economic concepts, elimination and substitution methods are often used to determine the company's Break Even condition, market equilibrium, National Income Break Even condition, etc. Company Break Even condition, which is a condition where the company's revenue (TR) is equal to all costs (TC). Market equilibrium, a condition where there is equality between demand (Dd) and market supply (Ss). Graphically, this condition occurs when the demand function line (Dd) intersects with the supply function line (Ss).