Solving systems of non-linear equations
Graphical solution of systems of nonlinear equations. The system of non-linear equations with two unknowns can be solved approximately using the graphs of these equations. The solution is the intersection of these graphs.
Consider the solution of a system of nonlinear equations in the following example.
Example 9.16. Find the solution of the system in the range of arguments (0.2; 3) in 0.2 increments.
Let's do the following:
• We represent the equations of the system in the form of functions: and
• Create the scope of the arguments and define the values of each of the functions at the points of this area.
• We construct graphs of functions for the equations of the system.
• Move the mouse pointer to the intersection point of the function graphs. The coordinates of the intersection point are displayed, which are approximate values of the solution of the system (Figure 9.33).
Fig. 9.33. Solving systems of non-linear equations
Solving systems of non-linear equations using the "Find Solution" tool. To solve non-linear equations systems, you can use the "Find Solution" tool.
Example 9.17. Consider the sequence of actions for solving systems of nonlinear equations for the following system:
Create a region for unknown variables and enter any values in the cells from the supposed domain of the system equations definition.
In the cells below, we write down the formulas for each of the system's functions: y 2 + x 2 < - 1 = 0 and ln x : + 2y = 0 , with references to the unknown variables described above.
Call the "Find Solution" tool on the screen, in the Optimize Target Function field, specify the address of the cell where the formula for one of the functions is written, set the switch to Value and write down the number 0.
In the Changing the cells of variables field, we will point to the cells in which unknown variables are written.
In the In accordance with the constraints we add constraints in the form of a link to the cell containing the formula of the second function, we limit it to 0 (Figure 9.34).
After finding the solution, we get x = 1, y = 0.
Fig. 9.34. Solving non-linear equation systems using the Solution search tool
This technology can also be used to solve systems of linear equations.
Determination of the equilibrium price. Determining the equilibrium price is always a dynamic process, which is the bargaining procedure. The seller assigns the price p 1 . For its part, the buyer, having evaluated the demand D strong> 1 at this price, assigns its price p 2 , at which the demand D 1 is equal to the sentence. For the sake of subjective reasons, the price p 2 is lower than equilibrium, as the buyer, for obvious reasons, tries to buy cheaper. In this situation, the seller estimates the demand D 2 , comparable to the price p 2 , and offers its price p 3, which in turn is above the equilibrium - the seller seeks to sell more. The development of this process of trading under certain conditions is steadily leading to an approximation to the equilibrium price.
In the economy, the known fact is the lower the price ( p ) , the higher the demand (D ) in conditions of the constant purchasing power of the population. As is known, most often the price-demand curve has the form of a descending line approaching the abscissa axis: D = c-ar, where c - the maximum demand for the goods; a - the dependence of the change in demand on price changes; p is the price of the product.
An increase in the price of a product leads to an increase in supply, expressed in the formula S = d + < strong> bp, where d is the minimum volume of the product on the market; b - the dependence of the change in the proposed volume of goods on price changes; p is the price of the item. It is economically advantageous to create a situation where there is a balance of supply and demand. If we assume that the dependence of demand on prices is expressed by the formula D , and the dependence of the offer on the price by the formula S = z ( ρ ) , then the equilibrium condition is expressed by the equation f p ) = z ( p ) and corresponds to the intersection of the D and S curves. In this regard, the price at which this condition is met is called equilibrium.
Example 9.18. Let's examine a problem in which the dependence of the demand y on a particular product from the price x is expressed by the equation y = x 2 + 2, and the sentence z from the price x - the equation z = = 3/x + 4. It is required to find an equilibrium point on the interval [0.5; 3] in steps of 0.1
Create a range for the price in steps of 0.1 and calculate the demand and supply values at these points.
Construct the function graphs for the demand and supply curves.
Move the mouse cursor to the intersection point of the graphs, at which the values of supply and demand are very close, we will assume that at this point the price of the commodity is in equilibrium (Figure 9.35).
Fig. 9.35. Determination of the equilibrium price
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