Use Cramer's Rule to solve the given linear system. x+y=1 x-y=2

Name: Problem 57, Chapter 4: Linear Algebra: A Modern Introduction
Uploaded: 2022-04-09T08:20:20-08:00
Duration: 4 min 40 s
Channel: Sikandar Baig
Description: Use Cramer's Rule to solve the given linear system. x+y=1 x-y=2

step 1 have been given a simultaneous linear equation. Let me write the equation first. The equation is

Use Cramer's Rule to solve the given linear system. ...

Key Concepts

Linear Systems of Equations

A collection of one or more linear equations involving the same set of variables. These systems are solved to find the values of the variables that satisfy all equations simultaneously, playing a fundamental role in algebra and applied mathematics.

Coefficient Matrix and Determinants

The coefficient matrix is constructed from the coefficients of the variables in a system of linear equations. The determinant of this matrix provides essential information about the system, such as the existence and uniqueness of solutions; a nonzero determinant usually indicates a unique solution.

Cramer's Rule

Cramer's Rule is a method for solving a system of linear equations with as many equations as unknowns. It involves calculating the determinant of the coefficient matrix and replacing columns with the constant terms to form new matrices whose determinants can be used to directly solve for the values of the individual variables.

Transcript

00:01 Have been given a simultaneous linear equation.

00:02 Let me write the equation first.

00:06 The equation is with two variables.

00:08 So let me write down the first equation.

00:11 That is we have x.

00:14 First of all, we have x plus y.

00:20 This is equal to one.

00:24 This is our first equation.

00:26 In the second equation, we have x minus y is equal to two.

00:32 So these are two.

00:33 Equations and we have to solve these equations through kramer's rule all right first of all to we will find the determinant of matrix a so what is matrix a matrix a is the matrix of coefficients here the coefficient is 1 here we have again 1 with x we have again coefficient 1 but with y we have a coefficient minus 1 alright so we have to find the determinant of a here 1 by minus 1 is minus 1 and 1 multiply by 1 minus into 1 times 1 is 1.

01:13 So from here the determinant of a comes out to be minus 2.

01:18 All right, dear learners.

01:20 Now let's find the determinant of matrix a1b.

01:41 So what is matrix a1b? in the first column of the matrix a1b we have the constants.

01:50 The constants are 1 and 2 and in the second column we have the coefficients of these ys that is 1 and minus 1 so this comprises of matrix a 1b this is equal to 1 times minus 1 is minus 1 minus 2 times 1 is 2 so from here the determinant of matrix a 1 b comes out to be negative 3 so similarly we will find the determinant of matrix a to b...

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