Cramer's Rule is a usefull tool to solve a set of simultaneous equations. This tool uses Cramer's Rules and formulas to solve these equations by converting them into a matrix form. This matrix form maybe like Ax =B. Each value of the variable is solved by using the determinants. The value of the variable is calculated separately and directly. Let us consider an example. Suppose we have N number of linear equations have n number of unknowns and the value of det(A) is not zero. This system will have a unique solution. The value of unknown x is given by the formula
Xn = det(An)/ det(A).


In this method we use the coofficents of the linear equations and make a matrix A. This way it becomes very easy to solve complex collection of equations. These are many steps involved in solving sets of equations using Cramer's Rules. The 1st step is to read the problem and see if the given matrix is a square matrix or not. Then the determinant of the given matrix is calculated and check if the value of A is not zero. If these 2 points are true then we will be able to apply Cramer's Rules. The next step is to replaces the 1st column by the elements which are on the right hand side of the equation that gives us determinant D1. Then D2 is got by replacing 2nd column with the elements in the right. Then by D1/D we get the variable x and D2/D will give value of variable y. We keep repeating these steps for other unknown variables.


Trapezoidal Rule is a useful method to find the exact value of a given definite integral. This is a numerical method and based on Newton-Cotes formula. The Newton-cotes formula states that the exact value of the integral can be calculated as nth order polynomial. This nth order polynomial integral gives us the exact value of the function. Lets say the n is one, in that case Trapezoidal Rule says that linear polynomial’s area will be given as
ʃba f(x) dx = ( b - a ) [ { f(a) + f(b) } / 2]
Lets takeup an example of Trapezoidal Rule. Now solve ʃ10 xdx using 4 sub-intervals for comparing actual and estimated value. In this we will find the change of x by b-a/n. From the given problem we get, ( 1 – 0 ) / 4 = ¼ = 0.25. Now we find the integral replacing the values into the formula we get 0.3125. Then since the actual value is 0.5, calculated by finding integral, we can find the percentage error. This percentage error is found by substracting the estimated values from the actual, divide that by actual and times 100. This formula is applied and we get 37.5%.
This way both Cramer's Rules and Trapezoidal Rule and helpful tools to calculate or solve problems. This tool is very effective and also helps in find the solutions of typical problems.


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